Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations
aa r X i v : . [ m a t h - ph ] N ov Riemann-Hilbert Problem and Quantum Field Theory:Integrable Renormalization, Dyson-Schwinger Equations
Ali Shojaei-Fard
November 2010 .. We are here, because we have a dream.But this is not a reason and this is not a why.The dream itself is a means and it is not an end ... ii cknowledgments First, it is my pleasure to thank Prof. Dr. Matilde Marcolli because of her helpful ideas, advicesand also, her great scientific supports during the Ph.D. program specially, my thesis. Second,I would like to thank Prof. Vida Milani because of her valuable encouragements and scientificsupports during the graduate period. Third, I would also like to thank Prof. Kurusch Ebrahimi-Fard and Prof. Dominique Manchon because of helpful and useful discussions at MPI and ESI.I acknowledge with thanks the financial supports from Hausdorff Research Institute for Mathe-matics (HIM) for the program Geometry-Physics (May 2008 - August 2008), Max Planck Institutefor Mathematics (MPIM) as a member of IMPRS program (September 2008 - February 2009),Erwin Schrodinger International Institute for Mathematical Physics (ESI) for the program Num-ber Theory and Physics (March 2009 - April 2009) and also, partially research fellowship fromM.S.R.T. (May 2008 - September 2008).Finally, I would like to thank Institute for Studies in Theoretical Physics and Mathematics(IPM) for giving time to me for talks [94, 97]. iii
BSTRACT
Finding a practical formalism for eliminating ultraviolet divergences in quantum field theorywas developed on the basis of perturbation theory. Kreimer could interpret the combinatorics ofBPHZ perturbative renormalization based on a Hopf algebraic structure on Feynman diagrams. Itwas applied to determine an infinite dimensional complex Lie group connected with a renormaliz-able theory underlying minimal subtraction scheme in dimensional regularization.Practically, Connes and Kreimer reformulated the BPHZ method based on the extraction offinite values with respect to the Birkhoff factorization on elements of the mentioned Lie group andthen Connes and Marcolli provided a new geometric interpretation from physical information basedon the Riemann-Hilbert correspondence. Moreover, they could introduce a categorical configurationto describe renormalizable theories such that as the result, formulating a universal treatment is animportant contribution in this direction.This Hopf algebraic approach improved immediately in several fields. On the one hand, withattention to the multiplicativity of renormalization and the theory of Rota-Baxter algebras, Connes-Kreimer theory determined an attractive procedure to consider integrable systems underlying renor-malizable theories. On the other hand, people reconsidered Quantum Chromodynamics, QuantumElectrodynamics and (non-)abelian gauge theories such that in this process, giving a comprehen-sive interpretation from non-perturbative theory is known as one important expected challenge inthe Connes-Kreimer-Marcolli theory. With applying Hochschild cohomology theory, Kreimer in-troduced a new combinatorial formulation from Dyson-Schwinger equations where it provides aninteresting intelligent strategy to analyze non-perturbative situations. Furthermore, Connes andMarcolli could discover a meaningful connection between quantum field theory and theory of mixedTate motives.This work is written based on this Hopf algebraic modeling in the study of quantum field theo-ries. We focus on some essential anticipated questions around this formalism for instance theoryof integrable systems and non-perturbative theory.In the first purpose, we concentrate on the theory of quantum integrable systems underlyingthe Connes-Kreimer approach. We introduce a new family of Hamiltonian systems depended onthe perturbative renormalization process (i.e. renormalization and regularization prescriptions) inrenormalizable theories. It is observed that the renormalization group can determine an infinitedimensional integrable system such that this fact provides a link between this proposed class ofmotion integrals and renormalization flow. Moreover, with help of the integral renormalizationtheorems, we study motion integrals underlying Bogoliubv character and BCH series to obtain anew family of fixed point equations.In the second goal, we consider the combinatorics of Connes-Marcolli approach to provide aHall rooted tree type reformulation from one particular object in this theory namely, universal Hopfalgebra of renormalization H U . As the consequences, interesting relations between this Hopf algebraand some well-known combinatorial Hopf algebras are obtained and also, one can make a new Hallpolynomial representation from universal singular frame such that based on the universal nature ofthis special loop, one can expect a Hall tree type scattering formula for physical information suchas counterterms.In the third aim, with attention to the given rooted tree version of H U and by applying theConnes-Marcolli’s universal investigation, we are going to improve the notion of an intrinsic geo-metrical interpretation from non-perturbative theory. In this process, at the first step we considercombinatorial Dyson-Schwinger equations at the level of the universal Hopf algebra of renormal-ization. At the second step, with respect to factorization of Feynman diagrams into primitivecomponents, the universality of H U at the level of these equations is discussed. And finally, wefind a bridge between these equations and objects of the universal category of flat equi-singularvector bundles such that by this way, the universal property of this category at the level of theseequations will be observed. iv eywords. Combinatorial Hopf Algebras; Connes-Kreimer Renormalization Group; Connes-Kreimer-Marcolli Perturbative Renormalization; Dyson-Schwinger Equations; Hall Rooted Trees;Quantum Integrable Systems; Renormalizable Quantum Field Theory; Riemann-Hilbert Corre-spondence; Universal Hopf Algebra of Renormalization.
MSC 2000.
PACS.
Email Address: [email protected] v ontents Introduction Theory of Hopf algebras Elements of Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Combinatorial Hopf algebras Connes-Kreimer Hopf algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2
Rooted trees and (quasi-)symmetric functions . . . . . . . . . . . . . . . . . . . . . . 143.3
Incidence Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Connes-Kreimer theory of the perturbative renormalization Hopf algebra of Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2
Algebraic perturbative renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Integrable renormalization: Integrable quantum Hamiltonian systems based on theperturbative renormalization What is an integrable system? From finite dimension (geometric approach) to infinitedimension (algebraic approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2
Rota-Baxter type algebras: Nijenhuis algebras . . . . . . . . . . . . . . . . . . . . . . 375.3
Theory of quantum integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . 40vi.4
Infinite integrable quantum Hamiltonian systems on the basis of the renormalizationgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.5
Rosenberg’s strategy: The continuation of the standard process . . . . . . . . . . . . 515.6
Fixed point equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Connes-Marcolli approach Geometric nature of counterterms: Category of flat equi-singular connections . . . . . 576.2
The construction of a universal Tannakian category . . . . . . . . . . . . . . . . . . . 60 Universal Hopf algebra of renormalization Shuffle nature of H U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.2 Rooted tree version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.3
Universal singular frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Combinatorial Dyson-Schwinger equations and Connes-Marcolli universal treatment Hopf algebraic reformulation of quantum equations of motion based on Hochschildcohomology theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.2
Universal Hopf algebra of renormalization and factorization problem . . . . . . . . . . 868.3
Categorical configuration in the study of DSEs . . . . . . . . . . . . . . . . . . . . . . 92 Conclusion and future improvements Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.2
Other integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.3
Relation between motion integrals and DSEs . . . . . . . . . . . . . . . . . . . . . . . 979.4
More about universal DSEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97vii hapter 1
Introduction ... ” You see, one thing is that I can live with doubt anduncertainty and not knowing. I think it’s much more interestingto live not knowing than to have the answers that might be wrong ”,Richard Feynman ...
Quantum Field Theory (QFT) is the most important profound formulated manifestation inmodern physics for the description of occurrences at the smallest length scales with highest en-ergies. In fact, this mysterious theoretic hypothesis is the fundamental result of merging the twocrucial achievements in physics namely, Quantum Mechanics and Special Relativity such that itsessential target can be summarized in finding an unified interpretation from interactions betweenelementary particles. The development of QFT is done in several formalizations such that Wight-man’s axiomatic configuration (i.e. constructive QFT) together with Haag’s algebraic formulationin terms of von Neumann algebras are the most influenced efforts [45, 85, 107]. Perturbationapproach is also another successful and useful point of view to QFT. It is based on perturbativeexpansions related to Feynman graphs where in these expansions one can find some ill-definediterated Feynman integrals such that they should be removed in a physical sensible procedure.This problem can be considered with a well-known analytic algorithm namely, renormalization.[77, 85]The apparatus of renormalization developed in the perturbation theory led to attractive suc-cess in quantum field theory where it has in fact an analytic meaning. But Kreimer could findthe appearance of a combinatorial nature inside of this analytic method and he showed that onecan explain the renormalization procedure (as a recursive formalism for the elimination of (sub-)divergences from diagrams) based on one particular Hopf algebra structure H F G on Feynmandiagrams. The discovery of this smooth mathematical construction encapsulated in the renormal-ization process covered the lack of a modern practical mechanism for the principally descriptionof this technique and also, it introduced a new rich relationship between this essential physicalmethod and modern mathematics. It should be mentioned that although the Bogoliubov recursionperforms renormalization without using any Hopf algebra structure but when we go to the higherloop orders, the advantages of this Hopf algebraic reconstruction in computations will be observeddecisively. [5, 6, 28, 29, 49, 59, 60, 62, 105, 106]There are different approaches to renormalization. For instance, in the Bogoliubov method, therenormalization is done without regularization and counterterms but in the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) method, we work on dimensional regularization (as the regularizationscheme) and minimal subtraction (as the renormalization map). Originally, the regularization canparametrize ultraviolet divergences appearing in amplitudes to reduce them formally finite together1ith a special subtraction of ill-defined expressions (associated with physical principles). But thisprocedure determines some non-physical parameters and indeed, it changes the nature of Feynmanrules (identified by the given physical theory) to algebra morphisms from the renormalization Hopfalgebra H F G to the commutative algebra A dr of Laurent series with finite pole part in dimensionalregularization. It is obviously seen that this commutative algebra is characterized with the givenregularization method and it means that by changing the regularization scheme, we should workon its associated algebra. In general, interrelationship between Feynman diagrams and Feynmanintegrals can be described by the Feynman rules of the theory and Kreimer could interpret thisrules on the basis of characters of H F G . [19, 30]Soon thereafter, Connes and Kreimer introduced a new practical reformulation for the BPHZperturbative renormalization. They could associate an infinite dimensional complex Lie group G ( C ) to each renormalizable physical theory Φ and then they investigated that this BPHZ schemeis in fact an example of a general mathematical procedure namely, the extraction of finite valueson the basis of the Riemann-Hilbert problem in the sense that one can calculate some importantphysical information for instance counterterms and related renormalized values with applyingthe Birkhoff decomposition on elements of this Lie group. In other words, according to theirprogramme, based on the regularization scheme, the bare unrenormalizaed theory produces ameromorphic loop γ µ on C = ∂ ∆ with values in the space of characters (i.e. the Lie group G ( C )) where ∆ is an infinitesimal disk centered at z = 0. For each z ∈ C , γ µ ( z ) = φ z is calleddimensionally regularized Feynman rules character. It means that after application of the specialcharacter φ on an arbitrary Feynman diagram Γ, one can get an iterated Feynman integral suchthat regularized version of this character namely, φ z maps each Feynman diagram Γ to a Laurentseries U zµ (Γ) in z with finite pole part (i.e. regularized unrenormalized version of the relatedFeynman integral). Then they showed that the renormalized theory is just the evaluation at theinteger dimension D of space-time of the holomorphic positive part of the Birkhoff decompositionof γ µ . As the result, they could reformulate physical information such as counterterms and relatedrenormalized values, renormalization group and its infinitesimal generator (i.e. β − function) basedon components of this decomposition of γ µ with values in the group of formal diffeomorphisms ofthe space of coupling constants. [11, 12, 16, 19, 37]The existence and the uniqueness of the Connes-Kreimer’s Birkhoff decomposition are con-nected with the Rota-Baxter property of the chosen regularization and renormalization couple(i.e. multiplicativity of renormalization). This fact reports about some interesting relations be-tween the theory of Rota-Baxter type algebras and the Riemann-Hilbert correspondence suchthat as a consequence one can expect to study quantum integrable systems in this Hopf algebraiclanguage. [24, 25, 26, 87]Connes and Marcolli widely improved this mathematical machinery from perturbative renor-malization by giving a categorical algebro-geometric dictionary for the analyzing of physical infor-mation in the minimal subtraction scheme in dimensional regularization underlying the Riemann-Hilbert correspondence. According to their approach, the dimensional regularization parameter z ∈ ∆ determines a principal C ∗ − bundle B over the infinitesimal disk ∆ (i.e. p : ∆ × C ∗ −→ ∆ ).Letting B := B − p − { } and P := B × G ( C ). Each arbitrary equivalence class ω of flat connec-tions on P denotes a differential equation D γ = ω (i.e. D : G ( C ) −→ Ω ( g ( C )) , φ φ − dφ )such that it has a unique solution. Equi-singularity condition on ω describes the independency ofthe type of singularity of γ at z = 0 from sections of B . It means that for sections σ , σ of B , σ ∗ ( γ ) and σ ∗ ( γ ) have the same singularity at z = 0. Connes and Marcolli firstly could reformu-late components of the Birkhoff factorization of loops γ µ ∈ Loop ( G ( C ) , µ ) based on time-orderedexponential and elements of the Lie algebra g ( C ). Secondly, they found a bijective correspon-dence between minus parts of this kind of decomposition on elements in Loop ( G ( C ) , µ ) (whichdetermine counterterms) and elements in g ( C ) and finally, they proved that each element of thisLie algebra determines a class of flat equi-singular connections on P . As the conclusion, onecan see that this family of connections encode geometrically counterterms such that the indepen-2ency of counterterms from the mass parameter µ is equivalent to the equi-singularity conditionon connections.In addition, they showed that these classes of connections can play the role of objects of acategory E Φ such that it can be recovered by the category R G ∗ of finite dimensional representationsof the affine group scheme G ∗ . In the next step and in a general configuration, they introduced theuniversal category of flat equi-singular vector bundles E such that its universality comes from thisinteresting notion that for each renormalizable theory Φ, one can put its related category of flatequi-singular connections E Φ as a full subcategory in E . Because of the neutral Tannakian natureof this universal category, one important Hopf algebra can be determined from the procedure.That is universal Hopf algebra of renormalization H U . By this special Hopf algebra and itsassociated affine group scheme U ∗ , renormalization groups and counterterms of renormalizablephysical theories have universal and canonical lifts. Connes and Marcolli developed this strongmathematical treatment to the motivic Galois theory. [16, 17, 18, 19]The systematic extension of this Hopf algebraic modeling to different kinds of (local) quantumfield theories would be an attractive topic for people and we can observe the improvement ofthis aspect for example in the reformulation of Quantum Electrodynamics (QED) (i.e. describesthe interaction of charged particles such as electrons with photons), Quantum Chromodynamics(QCD) (i.e. describes the strong interaction between quarks and gluons) and quantum (non-)abelian gauge theories. [4, 57, 61, 84, 99, 100, 101, 104]Furthermore, finding a comprehensive description from non-perturbative theory based on theRiemann-Hilbert problem is also known as an important and interesting challenge in this Hopfalgebraic viewpoint. In [16] the authors suggest a procedure by the Birkhoff factorization and theeffective couplings of the (renormalized) perturbative theories. On the other hand, there is a spe-cific class of equations in physics for the analytic studying of non-perturbative situations, namelyDyson-Schwinger equations (DSEs). With attention to the combinatorial nature of the Hopf alge-bra of renormalization (as the guiding structure) and also, with the help of Hochschild cohomologytheory, Kreimer introduced a new significant combinatorial version from these equations. Work-ing on classification and also calculating explicit solutions for these equations eventually lead to aconstructive achievement for the much better understanding of non-perturbative theory. One canrefer to [5, 6, 56, 58, 59, 60] for more details and considerable advances. The study of this kind ofequations on different Hopf algebras of rooted trees can help us to improve our knowledge for theidentification of combinatorial nature of non-perturbative events and its reason comes back to thisessential note that Hopf algebras of renormalizable theories are representable with a well-knownHopf algebra on rooted trees, namely Connes-Kreimer Hopf algebra. Some interesting resultsabout the study of DSEs on rooted trees are collected in [32, 40, 41]. In conclusion, with combin-ing the perturbation theory, the combinatorics of renormalization, the geometry of dimensionalregularization, the Connes-Marcolli categorification method and combinatorial Dyson-Schwingerequations, people hope to find a perfect conceptual understanding of quantum field theory.Finally, let us sketch the outline of the present dissertation. Here we are going to improveour knowledge about the applications of this Hopf algebraic formalism in the study of quantumfield theory. This research is on the basis of two general purposes namely, the theory of quantumintegrable Hamiltonian systems with respect to the Connes-Kreimer framework [92, 95] and thegeometric interpretation of non-perturbative theory with respect to the Connes-Marcolli approachand Dyson-Schwinger equations [96]. At the basis of these scopes, after reviewing some preliminaryfacts about Hopf algebras (in the next section), in chapter three we familiar with the Connes-Kreimer Hopf algebra and some others important combinatorial Hopf algebra structures. Thefourth section contains an overview from the Hopf algebraic perturbative renormalization. Inchapter five we consider our new point of view to study quantum integrable systems underlyingthe Connes-Kreimer theory [92, 94, 95]. This work yields a deep conceptional relation betweenthe theory of Rota-Baxter algebras and the Riemann-Hilbert problem. In the sixth part we studythe geometry of dimensional regularization and its categorical results. The consideration of a3all rooted tree type representation from universal Hopf algebra of renormalization is done inchapter seven. It makes possible to study interesting relations between this special Hopf algebraand some important well-known combinatorial Hopf algebras which have essential roles to studythe combinatorics of renormalization. In addition, we extend this new formulation of H U to thelevel of its associated complex Lie group (as a motivic Galois group) and Lie algebra. Withattention to this procedure, we provide a Hall rooted tree representation from one important loopnamely, universal singular frame γ U and then we will discuss about the application of this newreconstruction of γ U in the study of physical information [93]. In the eight section, with noticeto the given rooted tree version of H U , we study combinatorial DSEs at the level of this specificHopf algebra. And also, with attention to this class of equations and factorization of Feynmandiagrams into primitive components, we show that how one can extend the universality of H U to the level of non-perturbative theory. At last, with extending the universality of the categoryof equi-singular vector bundles to the level of DSEs, we improve the geometric knowledge aboutthis important class of equations in modern physics such that this process implies a categoricalconfiguration in the study of DSEs. [91, 96, 97] 4 hapter 2 Theory of Hopf algebras
The concept of Hopf algebra was introduced based on the work of Hopf on manifolds and then itswidely applications in topology, representation theory, combinatorics, quantum groups and non-commutative geometry displayed the power of this structure in different branches of mathematics[1, 47, 90]. Indeed, it is important to know that Hopf algebras provide generalizations for grouptheory and Lie theory. As a well-known example, it can be seen that the dual of the universalenveloping algebra of a simple Lie algebra determines a Hopf algebra. Moreover, this powerfulmathematical construction can provide a new opportunity to find useful interrelationships betweenthe pure world of mathematics and some complicate techniques in modern physics such as per-turbative renormalization. Additionally, with the help of a special class of Hopf algebras namely,quantum groups, one can observe the developments of this theory in mathematical physics andtheoretical physics [46, 48, 75, 80, 105].Since Hopf algebras play a skeleton key for this work, therefore it is essential to have enoughinformation about them. With attention to our future requirements, in this chapter we familiarwith the concept of Hopf algebra and then we will have a short overview from its basic properties.
Elements of Hopf algebras
Let K be a field with characteristic zero. A K − vector space A together with an associative bilinearmap m : A ⊗ A → A and a unit is called unital algebra such that its associativity and its unitare expressed respectively by the following commutative diagrams: A ⊗ A ⊗ A id ⊗ m (cid:15) (cid:15) m ⊗ id / / A ⊗ A m (cid:15) (cid:15) A ⊗ A m / / A (2.1.1) K ⊗ A µ ⊗ id / / ∼ % % KKKKKKKKKK A ⊗ A m (cid:15) (cid:15) A ⊗ K id ⊗ µ o o ∼ y y ssssssssss A (2.1.2)where the map µ : K −→ A is defined by µ ( λ ) = λ . This algebra is commutative , if m ◦ τ = m such that τ : A ⊗ A → A ⊗ A is the flip map defined by τ ( a ⊗ b ) = b ⊗ a. (2.1.3)5ith reversing all arrows in the above diagrams, one can define the dual structure of algebra.A K − vector space B together with a co-associative bilinear map ∆ : B → B ⊗ B and a counit ε : B −→ K is called coalgebra , if we have the following commutative diagrams. B ⊗ B ⊗ B B ⊗ B ∆ ⊗ id o o B ⊗ B id ⊗ ∆ O O B ∆ o o ∆ O O (2.1.4) K ⊗ B B ⊗ B ε ⊗ id o o id ⊗ ε / / B ⊗ K B ∆ O O ∼ e e KKKKKKKKKK ∼ ssssssssss (2.1.5)This coalgebra is co-commutative , if τ ◦ ∆ = ∆. With using the Sweedler’s notation for thecoproduct namely, ∆ x = X ( x ) x ⊗ x , (2.1.6)the co-associativity and the co-commutativity conditions will be written with(∆ ⊗ id ) ◦ ∆( x ) = X ( x ) x , ⊗ x , ⊗ x = X ( x ) x ⊗ x , ⊗ x , = ( id ⊗ ∆) ◦ ∆( x ) , (2.1.7) X ( x ) x ⊗ x = X ( x ) x ⊗ x . (2.1.8) Remark 2.1.1.
It is clear that sub-structures (i.e. sub-algebra, sub-coalgebra) are defined in anatural way.
For given algebra A and coalgebra B , one can define a product namely, convolution product onthe space L ( B, A ) of all linear maps from B to A . For each ϕ, ψ in L ( B, A ), it is given by ϕ ∗ ψ := m A ◦ ( ϕ ⊗ ψ ) ◦ ∆ B . (2.1.9) Definition 2.1.2. A K − vector space H together with the unital algebra structure ( m, µ ) and thecounital coalgebra structure (∆ , ε ) is called bialgebra, if ∆ and ε are algebra morphisms and µ is acoalgebra morphism. These conditions are determined with the following commutative diagrams. H ⊗ H ⊗ H ⊗ H τ / / H ⊗ H ⊗ H ⊗ H m ⊗ m (cid:15) (cid:15) H ⊗ H ∆ ⊗ ∆ O O m / / H ∆ / / H ⊗ H (2.1.10) H ⊗ H m (cid:15) (cid:15) ε ⊗ ε / / K ⊗ K ∼ (cid:15) (cid:15) H ⊗ H K ⊗ K µ ⊗ µ o o H ε / / K H ∆ O O K µ o o ∼ O O (2.1.11)6 efinition 2.1.3. A bialgebra H together with a linear map S : H −→ H is called Hopf algebra,if there is a compatibility between S and bi-algebraic structure given by the following commutativediagram. H ⊗ H S ⊗ id / / H ⊗ H m GGGGGGGGG H ε / / ∆ GGGGGGGGG ∆ ; ; K µ / / HH ⊗ H id ⊗ S / / H ⊗ H m ; ; (2.1.12) The map S is called antipode. There are many examples of Hopf algebras and with attention to our scopes in continue atfirst we familiar with some important Hopf algebras and reader can find more other samples in[1, 46, 90].
Example 2.1.4.
For a fixed invertible element q ∈ K , Hopf algebra H q is generated by andelements a, b, b − together with relations bb − = b − b = 1 , ba = qab. Its structures are determined by ∆ a = a ⊗ b ⊗ a, ∆ b = b ⊗ b, ∆ b − = b − ⊗ b − ,ε ( a ) = 0 , ε ( b ) = ε ( b − ) = 1 , S ( a ) = − b − a, S ( b ) = b − , S ( b − ) = b. [75] Example 2.1.5.
Let G be a finite group and K G the vector space generated by G . Identify a Hopfalgebra structure on K G such that its product is given by the group structure of G and e, ∆( g ) = g ⊗ g, ε ( g ) = 1 , S ( g ) = g − . [75] Example 2.1.6.
Let g be a finite dimensional Lie algebra over K . The universal envelopingalgebra U ( g ) is the noncommutative algebra generated by and elements of the Lie algebra withrespect to the relation [ x, y ] = xy − yx. It introduces a Hopf algebra structure such that ∆( x ) = x ⊗ ⊗ x, ε ( x ) = 0 , S ( x ) = − x. One can show that U ( g ) is the quotient of the tensor algebra T ( g ) modulo an ideal generated bythe commutator. [19] Example 2.1.7.
Consider the algebra of multiple semigroup H of natural positive integers N suchthat ( e n ) n ∈ N is its basis as a vector space. With the help of decomposition of numbers into theprime factors, one can define a commutative cocommutative connected graded (with the number ofprime factors (including multiplicities)) Hopf algebra such that its coproduct and its antipode aredetermined by ∆( e p ··· p k ) = X I ∐ J = { ,...,k } e p I ⊗ e p J ,S ( e n ) = ( − | n | e n , where p I denotes the product of the primes p j , j ∈ I . [74]
7y adding some additional structures such as grading and filtration, one can apply Hopfalgebras in physics. For example it helps us to classify all Feynman diagrams with respect to loopnumbers or number of internal edges and it will be useful when we do renormalization.
Definition 2.1.8. (i) A Hopf algebra H over K is called graded, if it is a graded K − vector space H = L n ≥ H n such that H p .H q ⊂ H p + q , ∆( H n ) ⊂ M p + q = n H p ⊗ H q , S ( H n ) ⊂ H n . (ii) A connected filtered Hopf algebra H is a K -vector space together with an increasing Z + -filtration: H ⊂ H ⊂ · · · ⊂ H n ⊂ · · · , [ n H n = H such that H is one dimension and H p .H q ⊂ H p + q , ∆( H n ) ⊂ X p + q = n H p ⊗ H q , S ( H n ) ⊂ H n . [24, 74] Lemma 2.1.9.
A graded bialgebra determines an increasing filtration H n = L np =0 H p . [74] The convolution product together with a filtration structure determine an antipode on a bial-gebra.
Lemma 2.1.10.
Any connected filtered bialgebra H is a filtered Hopf algebra. Its antipode struc-ture is given by S ( x ) = X k ≥ ( µε − Id H ) ∗ k ( x ) . [74] For a fixed connected filtered bialgebra H and an algebra A , set a map e := µ A ◦ ε H such that e ( ) = A and e ( x ) = 0 for any x ∈ Ker ε H . It is easy to see that e plays the role of unit for theconvolution product ∗ on the set L ( H, A ). Set G := { ϕ ∈ L ( H, A ) , ϕ ( ) = A } , (2.1.13) g := { α ∈ L ( H, A ) , α ( ) = 0 } . (2.1.14)It will be shown that elements of these sets make possible to find a new prescription from physicalinformation of a quantum field theory. Theorem 2.1.1. (i) ( G, ∗ ) is a group such that for each ϕ ∈ G , its inverse is given by ϕ ∗− ( x ) = (cid:0) e − ( e − ϕ ) (cid:1) ∗− ( x ) = X k ≥ ( e − ϕ ) ∗ k ( x ) . (ii) g is a subalgebra of ( L ( H, A ) , ∗ ) such that commutator with respect to the convolutionproduct introduces a Lie algebra structure on it.(iii) G = e + g .(iv) For any x ∈ H n , the exponential map is defined by e ∗ α ( x ) = X k ≥ α ∗ k ( x ) k ! . t determines a bijection map from g onto G such that its inverse namely, the logarithmic map isgiven by Log (1 + α )( x ) = X k ≥ ( − k − k α ∗ k ( x ) . (v) The above sums have just finite terms. [24, 46, 74, 80] It is also interesting to know that the increasing filtration on H can identify a complete metricstructure on L ( H, A ). Set L n := { α ∈ L ( H, A ) , α | H n − = 0 } . (2.1.15)It can be seen that for each positive integer numbers p and q , L p ∗ L q ⊂ L p + q . (2.1.16)It gives a decreasing filtration on L ( H, A ) such that L = L ( H, A ) and L = g . For each element ϕ ∈ L ( H, A ), the value val ϕ is defined as the biggest integer k such that ϕ is in L k . The map d ( ϕ, ψ ) = 2 − val ( ϕ − ψ ) (2.1.17)gives us a complete metric on L ( H, A ).We close this section with introducing one important technique for constructing Hopf algebrasfrom Lie algebras namely, Milnor-Moore theorem. Example 2.1.6 leads to a closed relation betweenLie algebras and Hopf algebras but in general, it is impossible to reconstruct a Hopf algebra froma Lie algebra. By adding some conditions, one can find very interesting process to recover Hopfalgebras.
Definition 2.1.11. (i) An element p in the Hopf algebra H is called primitive, if ∆( p ) = p ⊗ ⊗ p. (ii) A graded Hopf algebra is called finite type, if each of the homogenous components H i are finitedimensional vector spaces. One should mark that if the graded Hopf algebra H (of finite type) is an infinite dimensionalvector space, its graded dual H ∗ = M n ≥ H ⋆n (2.1.18)is strictly contained in the space of linear functionals H ⋆ := L ( H, K ). Remark 2.1.12.
Let H be a commutative (cocommutative) connected graded finite type Hopfalgebra.(i) Ker ε ≃ L i> H i is an ideal in H . It is called augmentation ideal.(ii) A linear map f ∈ H ⋆ belongs to H ∗ if and only if f | H i = 0 , for each component H i butfor a finite number.(iii) If H is finite dimensional vector space, then H ∗ = H ⋆ .(iv) There is a cocommutative (commutative) connected graded Hopf algebra structure on H ∗ .[24, 46, 74, 80] For a given Hopf algebra H , an element x in the augmentation ideal is called indecomposable ,if it can not been written as a linear combination of products of elements in Ker ε . The set of allindecomposable elements is denoted by In ( H ). Theorem 2.1.2.
For a given connected, graded and finite type Hopf algebra H ,(i) There is a correspondence between the set of all primitive elements of H namely, P rim ( H ) and In ( H ∗ ) .(ii) There is a correspondence between In ( H ) and P rime ( H ∗ ) . [46, 80] Theorem 2.1.3.
Let H be a connected graded commutative finite type Hopf algebra. It can bereconstructed with the Lie subalgebra P rim ( H ) of L ( H, K ) and it means that H ≃ U ( P rim ( H )) ∗ .[19, 80] Two classes of elements in H ⋆ have particular roles namely, characters and infinitesimal char-acters . It is discussed by Kreimer that Feynman rules of a given quantum field theory canbe capsulated in characters where this ability provides a new reformulation from counterterms,renormalized values, elements of the renormalization group and its related infinitesimal generator( β − function). Definition 2.1.13. (i) An element f ∈ H ⋆ is called character, if f (1) = 1 and for each x, y ∈ H , f ( xy ) = f ( x ) f ( y ) . (ii) An element g ∈ H ⋆ is called derivation (or infinitesimal character), if for each x, y ∈ H , g ( xy ) = g ( x ) ε ( y ) + g ( y ) ε ( x ) . Remark 2.1.14.
It is important to note that each primitive element of the Hopf algebra H ∗ determines a derivation. hapter 3 Combinatorial Hopf algebras
Hopf algebra of renormalization is introduced on the set of all Feynman diagrams of a givenrenormalizable physical theory such that its structures completely related to the renormalizationprocess on these graphs. Since we want to have a general framework to consider perturbationtheory, so it is necessary to identify a Hopf algebra structure independent of physical theoriesand further, it is reasonable to provide a universal simplified toy model for this Hopf algebraicformalization to apply it in computations. Fortunately, investigation of the combinatorics of therenormalization can help us to find a solution for this problem. Indeed, Kreimer applied decoratedversion of non-planar rooted trees (as combinatorial objects) to represent Feynman diagrams suchthat labels could help us to restore divergent sub-diagrams and their positions (i.e. nested loops)in origin graphs. Then with respect to the recursive mechanism for removing sub-divergences, heintroduced a coproduct structure on these labeled rooted trees such that as the result one canproduce a combinatorial Hopf algebra independent of physical theories. It is called Connes-KreimerHopf algebra of rooted trees [49, 55]. Even more in a categorical configuration, this rooted treetype model is equipped with a universal property with respect to Hochschild cohomology theorysuch that the grafting operator can determine its related Hochschild one cocycles [25, 26, 29]. Atthis level one can expect the applications of combinatorial techniques in the study of perturbativerenormalization [14, 28, 34, 40, 41, 42, 43, 44, 47, 68].In this chapter we focus on combinatorial objects and review the structures of some importantdefined combinatorial Hopf algebras which are connected with the Connes-Kreimer Hopf algeba.
Connes-Kreimer Hopf algebra
Rooted trees allow us to investigate the combinatorial basement of renormalization programmesuch that as one expected result, it determines a universal simplification for explaining the remov-ing of sub-divergences procedure. Here we consider the most important Hopf algebra structure onnon-planar rooted trees in the study of QFTs.
Definition 3.1.1.
A non-planar rooted tree t is an oriented, connected and simply connectedgraph together with one distinguished vertex with no incoming edge namely, root. A monomial inrooted trees (that commuting with each other) is called forest. •• ••• • /////// • (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) • •••• •• /////// • • (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) (3.1.1)A rooted tree t with a given embedding in the plane is called planar rooted tree. For example, •• ????????? • ????????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) •• ≇ •• ????????? •• (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • ????????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • (3.1.2)Let T be the set of all non-planar rooted trees and K T be the vector space over the field K (with characteristic zero) generated by T . It is graded by the number of non-root vertices ofrooted trees and it means that T n := { t ∈ T : | t | = n + 1 } , K T := M n ≥ K T n . (3.1.3)Consider graded free unital commutative symmetric algebra H ( T ) containing K T such that theempty tree is its unit. We equip this space with the counit ǫ : H ( T ) −→ K given by ǫ ( I ) = 1 , ǫ ( t ...t n ) = 0 , t ...t n = I . (3.1.4)With respect to the BPHZ renormalization process, some edges and vertices from rooted treesshould be removed step by step and this can be formulated with a special family of cuts. Definition 3.1.2.
An admissible cut c of a rooted tree t is a collection of its edges with thiscondition that along any path from the root to the other vertices, it meets at most one element of c . By removing the elements of an admissible cut c from a rooted tree t , we will have a rooted tree R c ( t ) with the original root and a forest P c ( t ) of rooted trees. For instance, • • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) •• •••
JJJJJJ • 7−→ R c ( t ) : •• ????? •• (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • P c ( t ) : • •• (3.1.5)shows an admissible cut but the cut • • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) •• •••• (3.1.6)12an not be admissible. This concept determines a coproduct structure on H ( T ) given by∆ : H ( T ) −→ H ( T ) ⊗ H ( T ) , ∆( t ) = t ⊗ I + I ⊗ t + X c P c ( t ) ⊗ R c ( t ) (3.1.7)where the sum is over all possible non-trivial admissible cuts on t . As an example,∆( ••• (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • ????? • ) = ••• ????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • ⊗ I + I ⊗ ••• ????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • + • ⊗ •• ????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • + •• ⊗ • ????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • +2 •••• ⊗ • + ••• ⊗ • • (3.1.8)It should be remarked that this coproduct can be rewritten in a recursive way. Let B + : H ( T ) −→ H ( T ) be a linear operator that mapping a forest to a rooted tree by connecting theroots of rooted trees in the forest to a new root. r • ttttttttttt(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? t • t • t n • (3.1.9)Operator B + is an isomorphism of graded vector spaces and for the rooted tree t = B + ( t ...t n ),we have ∆ B + ( t ...t n ) = t ⊗ I + ( id ⊗ B + )∆( t ...t n ) . (3.1.10)∆ is extended linearity to define it as an algebra homomorphism. On the other hand, with thehelp of admissible cuts one can define recursively an antipode on H ( T ) given by S ( t ) = − t − X c S ( P c ( t )) R c ( t ) . (3.1.11) Theorem 3.1.1.
The symmetric algebra H ( T ) together with the coproduct (3.1.7) and the an-tipode (3.1.11) is a finite type connected graded commutative noncocommutative Hopf algebra. Itis called Connes-Kreimer Hopf algebra and denoted by H CK . [10, 24, 25, 26] The study of Hopf subalgebras of H CK can be useful. For instance one can consider the cocom-mutative Hopf subalgebra of ladder trees (i.e. rooted trees without any side-branchings) H ( LT )such that it is applied to work on the relations between perturbative QFTs and representationtheory of Lie algebras. For this case, H ( LT ) is reduced to a polynomial algebra freely generatedby ladder trees and with the help of increasing or decreasing the degree of generators, one caninduce insertion and elimination operators. [78, 79]By theorem 2.1.1, the convolution product ∗ determines a group structure on the space charH CK of all characters and a graded Lie algebra structure on the space ∂ charH CK of allderivations where naturally, there is a bijection map exp ∗ from ∂ charH CK to charH CK (whichplays an essential role in the representation of components of the Birkhoff decomposition of char-acters). [25, 26]Finally one should mark to the universal property of this Hopf algebra such that it is theessential result of the universal problem in Hochschild cohomology.13 heorem 3.1.2. Let C be a category with objects ( H, L ) consisting of a commutative Hopf algebra H and a Hochschild one cocycle L : H −→ H . It means that for each x ∈ H , ∆ L ( x ) = L ( x ) ⊗ I + ( id ⊗ L )∆( x ) . And also Hopf algebra homomorphisms, that commute with cocycles, are morphisms in this cate-gory. ( H CK , B + ) is the universal element in C . In other words, for each object ( H, L ) there existsa unique morphism of Hopf algebras φ : H CK −→ H such that L ◦ φ = φ ◦ B + . H CK is unique upto isomorphism. [10] Rooted trees and (quasi-)symmetric functions
There are different Hopf algebra structures on (non-)planar rooted trees and in fact, Connes-Kreimer Hopf algebra is one particular choice. Here we try to familiar with some importantcombinatorial Hopf algebras and then with using (quasi-)symmetric functions, their relations with H CK will be considered. Definition 3.2.1.
Let t, s be rooted trees such that t = B + ( t , ...t n ) and | s | = m . The new product t (cid:13) s is defined with the sum of rooted trees given by attaching each of t i to a vertex of s . One can define a coproduct compatible with (cid:13) on K T . It is given by∆ GL B + ( t , ...t k ) = X I ∪ J = { , ,...,k } B + ( t ( I )) ⊗ B + ( t ( J )) . (3.2.1) Theorem 3.2.1. H GL := ( K T , (cid:13) , ∆ GL ) is a connected graded noncommutative cocommutativeHopf algebra and it is called Grossman-Larson Hopf algebra. H GL is the graded dual of H CK andit is the universal enveloping algebra of its Lie algebra of primitives. [40, 81] Let P be the set of all planar rooted trees and K P be its graded vector space. Tensor algebra T ( K P ) is an algebra of ordered forests of planar rooted trees and B + : T ( K P ) −→ K P is anisomorphism of graded vector spaces. There are two interesting Hopf algebra structures on P . Definition 3.2.2.
A balanced bracket representation (BBR) of a planar rooted tree contains sym-bols < and > satisfying in the following rules:- For a planar rooted tree of weight n , the symbol < and the symbol > occur n times,- In reading from left to right, the count of < ’s is agree with the count of > ’s,- The empty BBR is a tree with just one vertex. For example, one represents planar rooted trees ••• • /////// • (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) • •••• •• /////// • • (cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15)(cid:15) • ????????? • • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • (3.2.2)with <<>>, <><>, <<<>>>, <<><>>, <><><>, (3.2.3)respectively. 14 efinition 3.2.3. A BBR F is called irreducible, if F = < G > for some BBR G and otherwise itcan be written by a juxtaposition F ...F k of irreducible BBRs. These components correspond withthe branches of the root in the associated planar rooted tree. Definition 3.2.4.
Let t, s be two planar rooted trees with BBR representations F t , F s such that F t = F t ...F kt . Define a new product t ⋄ s by a sum of planar rooted trees such that their BBRs aregiven by shuffling the components of F t into the F s . Moreover, with help of the decomposition of elements into their irreducible components, onecan modify a compatible coproduct ∆ ⋄ on K P . Theorem 3.2.2. (i) Based on the balanced bracket representation, there is a connected gradednoncommutative Hopf algebra structure on K P and it is denoted by H P := ( K P , ⋄ , ∆ ⋄ ) .(ii) Based on the coproduct (3.1.7), there is a graded connected noncommutative Hopf algebra.It is called Foissy Hopf algebra and denoted by H F . H F is self-dual and isomorphic to H P .[32, 41, 44] Relation between rooted trees and noncommutative geometry can be clear when the recon-struction of one important Hopf algebra in computations of transverse index theory, based onrooted trees, is done [10, 47]. Consider a Hopf algebra H with the generators x, y, δ n ( n ∈ N ),together with the following relations[ x, y ] = − x, [ x, δ n ] = δ n +1 , [ y, δ n ] = nδ n , [ δ n , δ m ] = 0 , (3.2.4)such that its coproduct structure on the generators is given by∆( x ) = x ⊗ ⊗ x + δ , ∆( y ) = y ⊗ ⊗ y, ∆( δ ) = δ ⊗ ⊗ δ . (3.2.5)It is easy to present the generators δ n with rooted trees. Define a linear operator N on rootedtrees such that its application on a rooted tree t is a sum of rooted trees given by adding an edgeto each vertex of t . Now identify δ with • and δ n = N n ( I ). Lemma 3.2.5.
The set of generators { δ n } n introduces a Hopf subalgebra of H such that it isequivalent to the Connes-Moscovici Hopf algebra H CM . [10] It will be shown that how one can reduce H CM based on Dyson-Schwinger equations andadditionally, a copy of this Hopf algebra related to the universal Hopf algebra of renormalizationwill be determined. These results report the importance of this Hopf algebra in the study ofquantum field theory. Definition 3.2.6.
Let K [[ x , x , ... ]] be the ring of formal power series. A formal series f is called(i) symmetric, if for any sequence of distinct positive integers n , ..., n k , the coefficients in f of the monomials x i n ...x i k n k and x i ...x i k k equal.(ii) quasi-symmetric, if for any increasing sequence n < ... < n k , the coefficients in f of themonomials x i n ...x i k n k and x i ...x i k k equal.(iii) Let SY M ( QSY M ) be the set of all symmetric (quasi-symmetric) functions. It is easy tosee that
SY M ⊂ QSY M . For better understanding, it can be seen that for each n the symmetric group S n acts on K [[ x , x , ... ]] by permuting the variables and a symmetric function is invariant under these actionsand it means that after each permutation coefficients of its monomials remain without any change.15 emma 3.2.7. (i) As a vector space, QSY M is generated by the monomial quasi-symmetricfunctions M I such that I = ( i , ..., i k ) and M I := P n Theorem 3.2.3. (i) There is a graded connected commutative cocommutative self-dual Hopf al-gebra structure on SY M .(ii) There is a graded connected commutative non-cocommutative Hopf algebra structure on QSY M such that its graded dual is denoted by N SY M . As an algebra, N SY M is the noncom-mutative polynomials on the variables z n of degree n . [38, 41] Hoffman could find new important relations between rooted trees and (quasi-)symmetric func-tions such that we will extend his results to the level of the universal Hopf algebra of renormal-ization. Theorem 3.2.4. There are following commutative diagrams of Hopf algebra homomorphisms.[41] N SY M α −−−−→ H Fα y α y SY M α −−−−→ H CK SY M α ⋆ ←−−−− H GLα ⋆ y α ⋆ y QSY M α ⋆ ←−−−− H P (3.2.6) Proof. It is enough to define homomorphisms on generators. With attention to the definitions ofHopf algebras, we have- α sends each variable z n to the ladder tree l n of degree n .- α maps each planar rooted tree to its corresponding rooted tree without notice to the orderin products.- α sends each z n to the symmetric function m (1 , ..., | {z } n .- α maps m (1 , ..., | {z } n to the ladder tree l n .- For the composition I = ( i , ..., i k ) define a planar rooted tree t I := B + ( l i , ..., l i k ). For eachplanar rooted tree t , if t = t I , then define α ⋆ ( t ) := M I and otherwise α ⋆ ( t ) := 0.- For each rooted tree t , α ⋆ ( t ) := | sym ( t ) | P s ∈ α − ( t ) s .- α ⋆ is the inclusion map.- For the partition J = ( j , ..., j k ), define a rooted tree t J := B + ( l j , ..., l j k ). For each rootedtree t , if t = t J (for some partition J ), then define α ⋆ ( t ) := | sym ( t J ) | m J and otherwise α ⋆ ( t ) := 0. Definition 3.2.8. Recursively define the following morphism Z : N SY M −→ H GL , Z ( z n ) = ǫ n such that rooted trees ǫ n are given by ǫ := • ǫ n := k (cid:13) ǫ n − − k (cid:13) ǫ n − + ... + ( − n − k n where k n := X | t | = n +1 t | sym ( t ) | ∈ H GL . It is called Zhao’s homomorphism. Z is an injective homomorphism of Hopf algebras. Lemma 3.2.9. Dual of Zhao’s homomorphism exists uniquely. [42, 43]Proof. Suppose A + : QSY M −→ QSY M, M I M I ⊔ (1) . It is a linear map with the cocycle property. For each ladder tree l n and monomial u of rootedtrees, define a morphism Z ⋆ : H CK −→ QSY M such that l n M (1 , ..., | {z } n ,B + ( u ) A + ( Z ⋆ ( u )) . One can see that Z ⋆ is the unique homomorphism with respect to the map A + .We show that it is possible to lift the Zhao’s homomorphism and its dual to the level of Hallrooted trees and Lyndon words. Roughly, this process provides an extension of this homomorphismto the level of the universal Hopf algebra of renormalization. Incidence Hopf algebras On the one hand, rooted trees introduce one important class of Hopf algebras namely, combina-torial type and on the other hand, incidence Hopf algebras, induced in operad theory, provideanother general class of Hopf algebras such that rooted trees (as kind of posets) characterize inter-esting examples in this procedure. The essential part of this story is that incidence Hopf algebrarelated to one special family of operads introduces the Connes-Kreimer Hopf algebra.The story of operad theory was begun with the study of loop spaces and then its applications indifferent branches of mathematics were found very soon. There is a closed relation between operadsand objects of symmetric monoidal categories such as category of sets, category of topologicalspaces, category of vector spaces, ... . Additionally, operads can determine interesting source ofHopf algebras namely, incidence Hopf algebras [14, 86, 98]. In this part we are going to considerthis important family of Hopf algebras related to posets. Definition 3.3.1. A partially ordered set (poset) is a set with a partial order relation. A growingsequence of the elements of a poset is called chain. A poset is pure, if for any x ≤ y the maximalchains between x and y have the same length. A bounded and pure poset is called graded poset. Example 3.3.2. One can define a graded partial order on the set [ n ] = { , , ..., n } by the refine-ment of partitions and it is called partition poset. Definition 3.3.3. (i) An operad ( P, co , u ) is a monoid in the monoidal category S − M od of S − modules (i.e. a collection { P ( n ) } n of (right) S n − modules). It means that the compositionmorphism co : P ◦ P −→ P is associative and the morphism u : I −→ P is unit.(ii) This operad is called augmented, if there exists a morphism of operads ψ P : P −→ I suchthat ψ P ◦ u = id . Example 3.3.4. A S − set is a collection { P n } n of sets P n equipped with an action of the group S n . A monoid ( P, co , u ) in the monoidal category of S − sets is called a set operad. Definition 3.3.5. For a given set operad P and for each ( x , ..., x t ) ∈ P i × ... × P i t , define amap λ ( x ,...,x t ) : P t −→ P i + ... + i t , x co ( x ◦ ( x , ..., x t )) . A set opeard P is called basic, if each λ ( x ,...,x t ) be injective. Definition 3.3.6. Suppose ( P, co , u ) be a set operad and for the given set A with n elements, let A be the set of ordered sequences of the elements of A such that each element appearing once. Foreach n , there is an action of the group S n on P n such that for each element x n × ( a i , ..., a i n ) in P n × A , its image under an element σ of S n is given by σ ( x n ) × ( a σ − ( i ) , ..., a σ − ( i n ) ) . It is called diagonal action and its orbit is denoted by x n × ( a i , ..., a i n ) . Definition 3.3.7. Let P n ( A ) := P n × S n A be the set of all orbits under the diagonal action. Set P ( A ) := ( G f :[ n ] −→ A P n ) ∼ where f is a bijection and ( x n , f ) ∼ ( σ ( x n ) , f ◦ σ − ) is an equivalence relation. A P − partition of [ n ] is a set of components B , ..., B t such that- Each B j belongs to P i j ( I j ) where i + ... + i t = n ,- Family { I j } ≤ j ≤ t is a partition of [ n ] . Lemma 3.3.8. One can extend maps λ ( x ,...,x t ) to λ ∼ at the level of P ( A ) . [98]Proof. Define λ ∼ : P t × ( P i ( I ) × ... × P i t ( I t )) −→ P i + ... + i t ( A ) x × ( c , ..., c t ) co ( x ◦ ( x , ..., x t )) × ( a , ..., a ti t )such that { I j } ≤ j ≤ t is a partition of A and each c r is represented by x r × ( a r , ..., a ri r ) where x r ∈ P i r , I r = { a r , ..., a ri r } . Definition 3.3.9. For the set operad P and P − partitions B = { B , ..., B r } , C = { C , ..., C s } of [ n ] such that B k ∈ P i k ( I k ) and C l ∈ P j l ( J l ) , we say that the P − partition C is larger than B , iffor any k ∈ { , , ..., r } there exists { p , ..., p t } ⊂ { , , ..., s } such that- Family { J p , ..., J p t } is a partition of I k ,- There exists an element x t ∈ P t such that B k = λ ∼ ( x t × ( C p , ..., C p t )) .This poset is called operadic partition poset associated to the operad P and denoted by Π P ([ n ]) . One can develop the notion of this poset to each locally finite set A = F A n such that in thiscase a P − partition of [ A ] is a disjoint union (composition) of P − partitions of [ A n ]s and thereforethe operadic partition poset associated to the operad P will be a composition of posets Π P ([ A n ])and denoted by Π P ([ A ]). Definition 3.3.10. A collection ( p i ) i ∈ I of posets is called good collection, if- Each poset p i has a minimal element and a maximal element (an interval),- For all i ∈ I, x ∈ p i , the interval [ , x ] (or [ x, ] ) is isomorphic to a product of posets Q j p j (or Q k p k ). Remark 3.3.11. For a given good collection A := ( p i ) i ∈ I , it is possible to make a new goodcollection A − of all finite products Q i p i of elements such that it is closed under products andclosed under taking subintervals. [14] Let [ A ] ([ A − ]) be the set of isomorphism classes of posets in A ( A − ) such that elements inthese sets denoted by [ i ] , [ j ] , ... and H A be a vector space generated by the family { F [ i ] } [ i ] ∈ [ A − ] . Itis equipped with a commutative product (i.e. direct product of posets) F [ i ] F [ j ] = F [ i × j ] such that F [ e ] is the unit (where [ e ] is the isomorphism class of the singleton interval).18 emark 3.3.12. As an algebra H A may not be free. Lemma 3.3.13. Based on subintervals, there is a coproduct structure on H A given by ∆( F [ i ] ) = X x ∈ p i F [ ,x ] ⊗ F [ x, ] . It determines a commutative Hopf algebra. Theorem 3.3.1. Let Π P be a family of the operadic partition posets associated to the set operad P . One can find a good collection of posets ( p i ) (depended upon Π P ) such that its related Hopfalgebra H P is called incidence Hopf algebra. [14, 86] Remark 3.3.14. Incidence Hopf algebra H P has a basis indexed by the isomorphism classes ofintervals in the posets Π P ( I ) (for all sets I ) and this identification makes the sets I disappear andit means that the construction of this Hopf algebra is independent of any label. A rooted tree looks like a poset with a unique minimal element (root) such that for any element v , the set of elements descending v forms a chain (i.e. the graph has no loop) and maximal elementsare called leaves . There is an interesting basic set operad on rooted trees such that its incidenceHopf algebra determines a well known object. Definition 3.3.15. For the set I with the partition { J i } i ≥ , suppose N AP ( I ) be the set of rootedtrees with vertices labeled by I . For s i ∈ N AP ( J i ) and t ∈ N AP ( I ) , we consider the disjointunion of the rooted trees s i such that for each edge of t between i , i in I , add an edge betweenthe root of s i and the root of s i . The resulting graph is a rooted tree labeled by F i J i and its rootis the root of s k such that k is the label of the root of t . It defines the composition t (( s i ) i ∈ I ) . Theorem 3.3.2. Operad N AP is a functor from the groupoid of sets to the category of sets.[14, 86, 98] The operadic partition poset Π NAP ( I ) is a set of forests of I − labeled rooted trees such that aforest X is covered by a forest Y , if Y is obtained from X by grafting the root of one componentof X to the root of another component of X . Or X is obtained from Y by removing an edgeincident to the root of one component of Y . Remark 3.3.16. Any interval in Π NAP ( I ) is a product of intervals of the form [ , t i ] such that t i ∈ N AP ( J i ) . If t = B + ( t , ..., t k ) , then the poset [ , t ] is isomorphic to the product of the posets [ , B + ( t i )] for i ∈ { , , ..., k } . Lemma 3.3.17. The incidence Hopf algebra H NAP is a free commutative algebra on unlabeledrooted trees of root-valence such that elements F [ t ] (where t is a rooted tree) form a basis at thevector space level. According to the theorem 3.1.2 and the structure of H NAP , one can obtain the next importantresult. Theorem 3.3.3. H NAP is isomorphic to H CK by the unique Hopf algebra isomorphism ρ : F [ B + ( t ,...,t k )] t ...t k . [14] This theorem allows us to discover an operadic partition poset formalism for the Connes-Kreimer Hopf algebra of rooted trees such that after finding a rooted tree reformulation for H U ,one can apply theorem 3.3.3 to recognize an operadic source for this specific universal Hopf algebrain Connes-Marcolli treatment. 19 hapter 4 Connes-Kreimer theory of theperturbative renormalization The initial motivation in collaboration between the theory of Hopf algebras and the perturba-tion theory in renormalizable QFT was determined carefully with the description of perturba-tive renormalization underlying dimensional regularization in minimal subtraction scheme in analgebro-geometric framework. In other words, Connes and Kreimer discovered an interesting rev-olutionary bridge between the BPHZ prescription in renormalization and the Riemann-Hilbertcorrespondence. They proved that perturbative renormalization is in fact one special case ofthe general mathematical process of the extraction of finite values based on the Riemann-Hilbertproblem in the reconstruction of differential equations from data of their monodromy represen-tation such that for the algebraic reformalization of the BPHZ method, one can look at to thelocal regular-singular version of this problem where at this level the application of the Birkhofffactorization in the study of QFTs can be investigated. Because in fact, negative part of thisdecomposition can be applied to correct the behavior of solutions near singularities without intro-ducing new singularities. [11, 12, 17, 18, 19, 37]According to this mathematical mechanism, for a given renormalizable QFT Φ one can asso-ciate an infinite dimensional complex Lie group G ( C ) (i.e. Lie group of diffeographisms of thetheory) determined with the Hopf algebra H F G of Feynman diagrams of the theory and dependedon the chosen regularization method (i.e. a commutative algebra A ). It should be noticed thatsince Hopf algebra H F G is graded and finite type therefore the group G ( C ) is pro-unipotent.With working on the dimensional regularization in the minimal subtraction scheme, one canfind an algebro-geometric machinery to consider perturbative renormalization. It means that eachcharacter carries a geometric meaning in the sense that instead of working on characters one canreproduce physical information of a given theory Φ from factorization of loops (which are dependedon the mass parameter µ and the dimensional regularization parameter z ) with values in the Liegroup G ( C ).For instance in [11, 12], authors show that passing from unrenormalized value to the renormal-ized value is equivalent to the replacement of a given loop z γ µ ( z ) ∈ G ( C ) on the infinitesimalpunctured disk ∆ ∗ (identified by the regularization parameter) with the value of its positive com-ponent of the Birkhoff decomposition at the critical integral dimension D . In addition, one canrecover the related counterterm from the negative part of this decomposition.These results strongly depend on this essential fact that each regularized unrenormalized value U zµ (Γ( p , ..., p n )) determines a loop γ µ ( z ) on ∆ ∗ around the origin and with values in G ( C ) suchthat with the minimal subtraction this unrenormalized value for different values z will be sub-tracted. [16, 17]In this chapter, with a pedagogical intention, we are going to consider the Connes-Kreimer20pproach to renormalization to provide enough knowledge about this new Hopf algebraic inter-pretation from physical information. We start with the definition of the Hopf algebra of Feynmandiagrams and then consider some of its properties such as grading structures, gluing operator, itsrooted tree type representation. Finally, Hopf algebraic renormalization will be studied. Hopf algebra of Feynman diagrams A renormalizable perturbative quantum field theory can be introduced based on a family of graphsnamely, Feynman diagrams which describe possible circumstances between different types of el-ementary particles. In these graphs vertices report interactions and edges indicate propagators.Here one can see some examples of different types of vertices and edges in 3-dimensional scalarfield theory, QCD, QED and Gravity: • M-M-M-M-M-M-M-M- (cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m qQqQqQqQqQqQqQqQ 1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17) qQqQqQqQqQqQqQqQ 1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17) M-M-M-M-M-M-M-M- (cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m /o/o/o/o/o/o/o • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? _?_?_?_?_?_?_?_?_?_? (cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127) • qQqQqQqQqQqQqQqQ 1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17)1(cid:17) M-M-M-M-M-M-M-M- (cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m(cid:13) m/o/o/o/o/o/o/o _?_?_?_?_?_?_?_?_?_?_? (cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127)(cid:31) (cid:127) (4.1.1)Consider a theory with the set R V (consists of all possible interactions) and the set R E (consistsof all propagators). Definition 4.1.1. A Feynman diagram Γ is an oriented graph that contains a finite set Γ ofvertices and a finite set Γ of edges such that- For each vertex v , its type is determined by the set f v := { e ∈ Γ : e ∩ v = ∅} . - The set Γ decomposes into two different subsets(Int) Γ int consists of all internal edges (i.e. an edge together with begin and end vertices),(Ext) Γ ext consists of all external edges (i.e. an edge with an open end).- Based on Feynman rules of a theory, all edges are labeled with physical parameters (i.e.momenta of particles).- If p , ..., p k are momenta of external edges, then P i p i = 0 . (conservation law) (cid:1) (cid:1) (4.1.2)A special class of these diagrams together with an algebraic operation (i.e. insertion) areenough to construct the whole theory. They are one particle irreducible (1PI) Feynman graphswithout any sub-divergences which play the role of building blocks for defining a mathematicalstructure (i.e. Hopf algebra). Definition 4.1.2. An n-particle irreducible (n-PI) graph is a Feynman diagram Γ with this prop-erty that upon removal of n internal edges, it is still connected. It is clear that for n ≥ , eachn-PI graph is a (n-1)-PI. p • >p + k Definition 4.1.3. For each arbitrary Feynman diagram Γ ,(i) res (Γ) is a new graph as the result of shrinking all of the internal edges and vertices of Γ into one vertex. The resulting graph consists of a vertex together with all of the external edges of Γ . (ii) For each Feynman subgraph γ of Γ , the graph Γ /γ is defined by shrinking γ into a vertex.The resulting diagram is called quotient graph. /o/o/o • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????????????? res /o/o/o • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? (4.1.4)Γ = • /o/o/o/o/o/o/o/o/o/o/o/o/o/o/o < • • < <> • /o/o/o γ = < • • < >~>~ !a!a < Γ γ = /o/o/o/o/o/o/o • • >< /o/o/o/o/o/o/o (4.1.5) Definition 4.1.4. Define a bilinear operation ⋆ on the set of 1PI graphs given by Γ ⋆ Γ := X Γ n (Γ , Γ ; Γ)Γ where the sum is over 1PI graphs Γ and n (Γ , Γ ; Γ) counts the number of ways that a subgraph Γ can be reduced to a point in Γ such that Γ is obtained and also | Γ | = | Γ | + | Γ | , res (Γ) = res (Γ ) .⋆ (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????? = (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? (cid:127)(cid:127)?? + (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? ??(cid:127)(cid:127) + (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? (cid:127)(cid:127)?? (4.1.6)22 (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????? ⋆ = 2 (4.1.7) Remark 4.1.5. (i) Finitely of Feynman diagrams show that the above sum is finite.(ii) res (Γ ⋆ Γ ) = res (Γ ) ,(iii) The operation ⋆ is pre-Lie, namely [Γ ⋆ Γ ] ⋆ Γ − Γ ⋆ [Γ ⋆ Γ ] = [Γ ⋆ Γ ] ⋆ Γ − Γ ⋆ [Γ ⋆ Γ ] , (iv) For some integers r and k j that j = 1 , ..., r , any non-primitive 1PI graph Γ can be writtenat most in r different forms Γ = k j Y i =1 γ j ⋆ j,i Γ j,i such that γ j s are primitive graphs. When r > , the graph Γ is called overlapping. [54, 59, 60] So this operator determines a Lie algebra structure on Feynman diagrams such that the Liebracket is the commutator with respect to the ⋆ . From physical point of view, this insertionoperator technically can be expounded by the gluing of Feynman graphs based on types of edges.The reader interested in this quest of a deeper level of understanding should consult [10, 13, 54]. Theorem 4.1.1. Graded dual of the universal enveloping algebra of the Lie algebra L on 1PIgraphs (determined with the definition 4.1.4) is a graded connected commutative non-cocommutativeHopf algebra. It is called Hopf algebra of Feynman diagrams of the theory Φ and denoted by H F G = H (Φ) . [16, 19, 49, 55]Proof. It is the immediate result of the Milnor-Moore theorem. Based on the gluing information,one can determine sub-diagrams of Feynman graphs such that it leads to the coproduct structure.For each Feynman diagram Γ, its coproduct can be written by∆(Γ) = Γ ⊗ I + I ⊗ Γ + X γ ⊂ Γ γ ⊗ Γ /γ such that the sum is over all disjoint unions of 1PI superficially divergent proper subgraphs withresidue in R V ∪ R E where the associated amplitudes of their residues need renormalization. Nowexpand it to the free products of 1PI graphs. Remark 4.1.6. There are different choices for grading structure on the Hopf algebra of Feynmandiagrams such as number of vertices, number of internal edges, number of independent loops, ....Grading with the number of internal edges determines finite type property for this Hopf algebra.[16, 19] It is remarkable to know that one can reformulate this Hopf algebra by a certain decoratedversion of the Connes-Kreimer Hopf algebra of rooted trees such that decorations conserve somephysical information such as (sub-)divergences (i.e. nested loops) of Feynman diagrams. As an23xample, the diagram • •• • • •• •• •• • • • • • (4.1.8)can be represented by the labeled rooted tree •• ????????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) • •• ••• (4.1.9)such that each vertex reports the divergent primitive sub-diagram • • and edges showthe locations of sub-diagrams in the main graph. Representation of Feynman diagrams togetherwith overlapping divergences based on rooted trees is also studied. In [50, 68] authors show thathow these kinds of nested sub-divergences can be reduced to linear combinations of rooted trees.The Lie algebra L gives rise to two representations acting as derivations on H (Φ). They are < Z +Γ , Γ > := Γ ⋆ Γ (4.1.10) < Z − Γ , Γ > := X i < Z +Γ , (Γ ) ′ i > (Γ ) ” i (4.1.11)such that ∆(Γ ) = I ⊗ Γ + Γ ⊗ I + X i (Γ ) ′ i ⊗ (Γ ) ” i . (4.1.12) Remark 4.1.7. If Γ be a 1PI graph, then for each term in the above sum, there is a uniquegluing data G i that describes how one can reach to the graph Γ by gluing of the components (Γ ) ′ i into (Γ ) ” i . [13, 34, 54] There is also another interesting and useful grading where it can be applied to explain therelation between the Connes-Kreimer Hopf algebra of rooted tress and the Hopf algebra of Feynmandiagrams, proof of locality of counterterms and also in the study of Dyson-Schwinger equations. Definition 4.1.8. Let Ker ǫ be the augmentation ideal of H (Φ) and P : H (Φ) −→ ker ǫ, P := id − I ǫ e the projection on to this ideal. Define a new map Aug ( m ) := ( P ⊗ ... ⊗ P | {z } m )∆ m − : H (Φ) −→ { ker ǫ } ⊗ m and set H (Φ) ( m ) := KerAug ( m +1) KerAug ( m ) , m ≥ . It is called bidegree. Lemma 4.1.9. One can show that H (Φ) = M m ≥ H (Φ) m = M m ≥ H (Φ) ( m ) such that for each m ≥ , H (Φ) m ⊂ m M j =1 H (Φ) ( j ) , H (Φ) ≃ H (Φ) (0) ≃ K . [59, 74] Remark 4.1.10. (i) All Feynman graphs that contain (sub-)divergences (i.e. nested loops) belongto the augmentation ideal and it means that H aug (Φ) := L i ≥ H (Φ) i stores quantum information.(ii) For each 1PI graph Γ , one can identify a linear generator δ Γ and set H lin (Φ) := span { δ Γ } Γ .It is observed that H lin (Φ) ⊂ H aug (Φ) . The grafting operator B + is defined on rooted trees but with attention to the decorations onecan lift it to the level of Feynman diagrams. For much better understanding of this translation,letting H CK (Φ) be a labeled version of the Connes-Kreimer Hopf algebra of rooted trees (decoratedby primitive 1PI Feynman graphs of the renormalizable theory Φ). By choosing a primitive element γ , the operator B + γ is an homogeneous operator of degree one such that after its application to aforest, it connects the roots in the forest to a new root decorated by γ . As an example, one cansee that B + > ( > ) = > (4.1.13)In chapter eight, it will be discussed that the grafting operator determines Hochchild one cocyclesof a chain complex connected with the renormalization coproduct. Now bidegree (as the gradingfactor) and the operator B + can provide a decomposition of diagrams which contain primitivecomponents such that it helps us to have a practical instruction for studying Feynman diagrams. Theorem 4.1.2. Define a homomorphism Ψ : H (Φ) −→ H CK (Φ) given by Ψ(Γ) = r X j =1 B + γ j ,G j,i [ k j Y i =1 Ψ(Γ j,i )] . One can show that(i) Ψ is defined by induction on bidegree,(ii) It is a Hopf algebra homomorphism,(iii) Its image is a closed Hopf subalgebra,(iv) G j,i s are the gluing data,(v) B + γ j ,G j,i s are one-cocycles. [13, 34, 54, 74] Theorem 4.1.3. For a given renormalizable quantum field theory Φ ,(i) The discrete Hopf algebra of Feynman diagrams H D (Φ) is made on the free commutativealgebra over C generated by pairs (Γ , w ) such that Γ is a 1PI graph and w is a monomial withdegree agree with the number of external edges of graph.(ii) The full Hopf algebra H F (Φ) is made on the symmetric algebra of the linear space ofdistributions defined by the external structures.(iii) H (Φ) is isomorphic to H D (Φ) (in the case: without external structure) or H F (Φ) (in thecase: with external structures). [11, 12, 19, 59] It is discussed that how one can arrange Feynman diagrams of given physical theory into aHopf algebra based on the recognizing of sub-divergences of diagrams. Now it is important to havean explicit understanding from the concept of ” renormalizability ” underlying this Hopf algebrastructure and so at the final part of this section we consider this essential concept.Start with the Largrangian of a given physical theory Φ. We know that each monomial in theLagrangian corresponds to an amplitude. Letting A be the set of all amplitudes. Definition 4.1.11. The physical theory Φ is renormalizable if it has a finite subset R + ⊂ A asthe set of amplitudes which need renormalization. For example, it can be easily seen that φ in dimension D ≤ a ∈ A specifies an integer n = n ( a ) which gives the number of externaledges. Let M a be the set of all 1PI graphs contributing to the amplitude a , | Γ | be the number ofindependent loops in Γ and | sym (Γ) | be the rank of the automorphism group of the graph. Definition 4.1.12. Let φ be the Feynman rules character associated to the theory Φ . The Greenfunction related to an amplitude a and the character φ is given by G aφ = 1 ± X Γ ∈M a α | Γ | φ (Γ) | sym (Γ) | = 1 ± X k ≥ α k φ ( c ak ) where c ak = X Γ ∈M a , | Γ | = k Γ | sym (Γ) | such that the sum is over all 1PI graphs of order k contributing to the amplitude a . The plus signis taken if n ( a ) ≥ and the minus sign for n ( a ) = 2 . Since we are interested to study graphs together with (sub-)divergences, therefore it is notnecessary to consider all graphs. Because for instance in φ theory in six dimension (as a toymodel), one investigates that superficial divergent diagrams are those with the number of externaledges ≤ tadpole amplitudes (i.e. n ( a ) = 1) and vacuum amplitudes (i.e. n ( a ) = 0)have vanished Green functions. • • V acuum • • T adpole (4.1.14)26ndeed, it is enough to study Green functions depended on 1PI graphs with 2 or 3 external legs.Because with the help of these graphs, one can build a basis for the Hopf algebra H F G . Now byindicating the structure of the sumΓ a = I ± X Γ ∈M a α | Γ | Γ | sym (Γ) | (4.1.15)in the definition 4.1.12, it is implicitly observed that after the application of the Feynman rulescharacter φ on this sum, we will obtain its related Green function and it means that G aφ = φ (Γ a ) . (4.1.16) Lemma 4.1.13. For a given renormalizable theory Φ , one can decompose the set of amplitudes A into two disjoint subsets R + , R − such that for each r ∈ R + , we have Γ r = I ± X k ≥ α k B + k ; r (Γ r Q n r k ) where B + k ; r are Hochschild 1-cocycles and Q n r is a monomial in Γ r or its inverse. Since theamplitudes from the set R − are determined with the knowledge of elements in R + , therefore for thestudy of renormalizable theories, it is enough to focus on the elements in R + . [5, 6, 57, 59, 60, 66] Recognizing Feynman diagrams together with (sub-)divergences can help us to consider moreprecisely their related Feynman integrals. There is an important factor (connected with thedimension of theory) to identify divergences in graphs namely, superficial degree of divergency ω . Remark 4.1.14. Parameter ω has the following properties:(i) All Feynman diagrams with the same external structure have the same superficial degree ofdivergence. It means that res (Γ ) = res (Γ ) = ⇒ ω (Γ ) = ω (Γ ) . (ii) This degree shows the existence of divergency only for a finite number of distinct externalstructures r in R + . [34] Actually, the superficial degree of divergency of a 1PI graph Γ measures the degree of singularityof the integral in the amplitude a Γ with respect to the integrated variables q , q , .... . Under thetransformation of momenta q i tq i ( t ∈ R ), we have a Γ t ω (Γ) a Γ . With notice to this factor,a classification of amplitudes of a given renormalizable theory is possible. Lemma 4.1.15. The amplitude of a diagram Γ with just one loop(i) is convergent if ω (Γ) < ,(ii) has a logarithmic divergency if ω (Γ) = 0 ,(iii) has a polynomial divergency if ω (Γ) > . The computation of the superficial degree of divergency of a Feynman graph Γ with more thanone loop is also possible with starting from 1PI subgraphs with one loop and continue the processby enlarging subgraphs until to reach to the main graph. Remark 4.1.16. If we look at to the above procedure, the existence of a self-similar recursive waydetermines the formal sums Γ r ( r ∈ R + ) in terms of themselves and the action of suitable maps(i.e. one cocycles) B + k ; r . For each Feynman diagram Γ, B + n (Γ) is defined by the insertion of a collection of (sub-)divergences Γ into the identified n − loop primitive graphs. Kreimer could introduce a new mea-sure to translate the combinatorics of Feynman diagrams to the normal analytic picture whichphysicists familiar with it. 27 emma 4.1.17. There is a measure µ + such that for each Feynman rules character φ , it is deter-mined by φ ( B + n ( I )) := R dµ + where the expression φ ( B + n (Γ)) = R φ (Γ) dµ + shows that subgraphsbecome subintegrals under the Feynman rules. [56, 61, 66] In summary, it is considered that for each arbitrary renormalizable perturbative physical theoryone can introduce a graded connected commutative non-cocommutative Hopf algebra of finite typesuch that 1PI Feynman graphs play the role of its building blocks. And also, with the help ofa decorated version of the Connes-Kreimer Hopf algebra of rooted trees (such that primitivedivergent sub-graphs are put in labels), one can display a toy model from this Hopf algebra whereit will be useful to do renormalization in a simpler procedure for different theories. This toy modelcan provide a universal framework in the study of different theories. Algebraic perturbative renormalization Attempts at eliminating divergences had been started from the birth of quantum field theory. Inexperimentally calculation of amplitudes, we have encountered two different kinds of divergences: infra-red (i.e. an amplitude which becomes infinite for vanishingly small values of some momenta)and ultra-violet (i.e. an amplitude which becomes infinite for arbitrary large values of momentain a loop integration). The applied structure in interpretation of ultra-violet divergences will alsoallow to cancel infra-red divergences. The main idea of renormalization is to correct the originalLagrangian of a quantum field theory by an infinite series of counterterms, labeled with Feynmangraphs of the theory. By these counterterms we can disappear ultra-violet divergences. [19, 69, 77]Feynman diagrams together with some special rules have the ability of presenting all of thepossible happenings in a renormalizable theory. For example in a Feynman graph, external edgesare the symbols for particles with assigned momenta, vertices show interactions and internal edgesfor creation and annihilation of virtual particles are applied.With applying Feynman rules of the given theory, one can associate a Feynman iterated integralto each Feynman diagram. Generally since these integrals have (sub-)divergences, therefore theyare complex and ill-defined. This problem is in fact the most conceptional difficulty in quantumfield theory such that with the help of various approaches, people try to find suitable solutionsfor this problem. As a consequence, nowadays renormalization is one understandable powerfultechnique to consider the behavior of diagrams together with (sub-)divergences. Furthermore,we studied that how one can connect a Hopf algebra structure to a given theory and now itsurely very important to know that how this new mathematical structure leads to a new algebraicreformulation from the perturbative renormalization process underlying BPHZ method. It shouldbe remarked that the wide advantage of this point of view (in the study of renormalization) thanthe Bogoliubov recursive standard process can be observed more clearly, when we want to workon the Feynman diagrams with high loop orders. [13, 28, 50, 59, 61, 62, 105, 106]Here we are going to have a short overview from renormalization process with respect to thisHopf algebraic modeling. This formalism is discovered by Connes and Kreimer and they couldfind very closed relation between this well-known physical technique and one important generalinstruction in mathematics, namely Riemann-Hilbert problem.There are several mechanisms for renormalization. Connes-Kreimer approach to the pertur-bative renormalization is certainly a practical reformulation from the BPHZ method on the basisof the Hopf algebra H F G connected with a given renormalizable theory Φ such that Feynmanrules can be determined with special characters of this Hopf algebra. One should point out thatbecause of the universal property of the Connes-Kreimer Hopf algebra H CK of rooted tree (withrespect to the Hochschild cohomology), the using of a decorated version of this Hopf algebra incomputations helps us to find a simplified toy model for studying. [25, 26]So we should concentrate on the renormalization procedure depended upon one regularizationparameter. In this kind, the first stage of renormalization is done by regularization such that with28he help of regularization parameter, all divergences appearing in amplitudes can be parameterizedto reach to the formally finite expansions together with a subtraction of ill-defined expressions. Inthe regularization process some non-physical parameters will be created such that this fact couldbe changed the nature of Feynman rules to algebra homomorphisms from the Hopf algebra H F G to a commutative algebra A where this algebra is characterized by the regularization prescription.Connes and Kreimer proved that minimal subtraction scheme in dimensional regularization canbe rewritten based on the Birkhoff decomposition of characters of H F G with values in the alge-bra A dr of Laurent series with finite pole part such that components of this factorization of veryspecial characters (identified with the Feynman rules of a given theory) provide counterterms andrenormalized values. Moreover, one can redefine the renormalization group and its infinitesimalgenerator ( β − function) by using the negative component. In this procedure, one important alge-braic property plays an essential role. It is the Rota-Baxter property of the pair ( A dr , R ms ) suchthat it reports the multiplicativity of renormalization and it leads to apply the Riemann-Hilbertcorrespondence in the study of perturbative QFT. [24, 25, 26, 105] Definition 4.2.1. Supposing that H be the Hopf algebra of Feynman diagrams of a renormalizableQFT Φ and A be a commutative algebra with respect to the regularization scheme. Set G ( A ) := Hom ( H, A ) = { φ : H −→ A : φ ( xy ) = φ ( x ) φ ( y ) , φ (1) = 1 } ⊂ L ( H, A ) and consider the convolution product on G ( A ) such that for each φ , φ ∈ G ( A ) , it is given by φ ∗ φ := m A ◦ ( φ ⊗ φ ) ◦ △ H . Theorem 4.2.1. (i) The convolution product ∗ determines a group structure on G ( A ) .(ii) For a fixed commutative Hopf algebra H , there is a representable covariant functor G (represented by H ) from the category of unital commutative K − algebras to the category of groups.(iii) Every covariant representable functor between two above categories is determined by anaffine group scheme G . For each commutative algebra A , G ( A ) is called affine group scheme.[11, 12, 15, 16, 17, 18, 19] One can improve this categorical configuration to the Lie algebra level. Theorem 4.2.2. For a given affine group scheme G (viewed as a functor), one can extend it tothe covariant functor g = Lie G , A g ( A ) from the category of commutative unital K − algebras to the category of Lie algebras. g ( A ) is theLie algebra of linear maps l : H −→ A such that for each x, y ∈ H , l ( xy ) = l ( x ) ǫ ( y ) + ǫ ( x ) l ( y ) where ǫ is the augmentation of H . [15, 16, 17, 18, 19] Remark 4.2.2. Equivalently there is another picture from the elements of g ( A ) given by linearmaps t : H −→ H with the properties t ( xy ) = xt ( y ) + t ( x ) y, △ t = ( Id ⊗ t ) △ . In this case, the Lie bracket is defined by the commutator with respect to the composition. [17, 19] Since characters include Feynman rules of a theory therefore it is important to have enoughinformation about the dual space L ( H, A ) of all linear maps from H to A . We know that filtrationcan determine a concept of distance related to Hopf algebras. Loop number of Feynman diagramsinduces an increasing filtration on H such that in the dual level, it defines a decreasing filtration29n L ( H, A ). With the help of (2.1.15), (2.1.16) and (2.1.17), one can have a complete metric d Φ on the space L ( H, A ). In future it will be seen that the dual space plays an essential role in thestudy of quantum integrable systems.It is good place to familiar with another additional algebraic structure namely, Rota-Baxtermaps. Definition 4.2.3. Let K be a field with characteristic zero and A an associative unital K − algebra.A K − linear map R : A −→ A is called Rota-Baxter operator of weight λ ∈ K , if for all x, y ∈ A ,it satisfies R ( x ) R ( y ) + λR ( xy ) = R ( R ( x ) y + xR ( y )) . (4.2.1) The pair ( A, R ) is called Rota-Baxter algebra. Lemma 4.2.4. (i) For λ = 0 , the standard form R ( x ) R ( y ) + R ( xy ) = R ( R ( x ) y + xR ( y )) is given by the transformation R λ − R .(ii) If R is a Rota-Baxter map, then e R := Id A − R will be a Rota-Baxter map and also Im R,Im e R are subalgebras in A .(iii) For a given Rota-Baxter algebra ( A, R ) , define a product a ◦ R b := R ( a ) b + aR ( b ) − ab on A . A R := ( A, ◦ R , R ) is a Rota-Baxter algebra. It is called double Rota-Baxter algebra and onecan continue this process to obtain an infinite sequence of doubles.(iv) The pair ( A, R ) has a unique Birkhoff decomposition ( R ( A ) , − e R ( A )) ⊂ A × A if and onlyif it is a Rota-Baxter algebra.(v) There is a natural way to extend the Rota-Baxter property to the Lie algebra level. For agiven Rota-Baxter algebra ( A, R ) , let ( A, [ ., . ]) be its Lie algebra with respect to the commutator.One can show that for each x, y ∈ A , [ R ( x ) , R ( y )] + R ([ x, y ]) = R ([ R ( x ) , y ] + [ x, R ( y )]) . (4.2.2) Triple ( A, [ ., . ] , R ) is called Lie Rota-Baxter algebra. [24, 25, 26, 74] Based on a given Rota-Baxter structure on the algebra A , it is possible to determine a newalgebraic structure on the dual space L ( H, A ) such that it will be applied to reformulate renor-malization process. Theorem 4.2.3. Suppose regularization and renormalization schemes in a given theory are in-troduced by Rota-Baxter algebra ( A, R ) . Another Rota-Baxter map Υ can be inherited on L ( H, A ) given by φ Υ( φ ) := R ◦ φ. (i) Triple e Φ := ( L ( H, A ) , ∗ , Υ) is a complete filtered noncommutative associative unital Rota-Baxter K − algebra of weight one.(ii) One can extend this Rota-Baxter map to the Lie algebra level. [24, 37] Now consider one particular renormalization method in modern physics namely, minimal sub-traction scheme in dimensional regularization. This interesting scheme is identified with the com-mutative algebra of Laurent series with finite pole part A dr := C [[ z, z − ] and the renormalizationmap R ms on A dr where it is given by R ms ( ∞ X i ≥− m c i z i ) := − X i ≥− m c i z i . (4.2.3)30t is an idempotent Rota-Baxter map of weight one. Connes and Kreimer proved that the Bo-goliubov’s recursive formula for counterterms and renormalized values (depended on determinedcharacters) can be reconstructed with the Birkhoff decomposition on elements of G ( A dr ) and themap R ms [11, 12, 19]. Theorem 4.2.4. For a given renormalizable physical theory Φ underlying the BPHZ method withthe related Hopf algebra H D (Φ) (for the discrete level) and H F (Φ) (for the full level),(i) The groups of diffeographisms Diffg (Φ) D (for the discrete level) and Diffg (Φ) F (for the fulllevel) are the pro-unipotent affine group schemes of the Hopf algebras H D (Φ) and H F (Φ) . Therelation between these two groups is determined by a semidirect product as follows Dif f g (Φ) F = Dif f g (Φ) ab ⋊ Dif f g (Φ) D . (ii) Set H = H (Φ) . The graded dual Hopf algebra H ∗ contains all finite linear combinationsof homogeneous linear maps on H . If L := Lie P rimH ∗ , then there is a canonical isomorphismof Hopf algebras between H and the graded dual of the universal enveloping algebra of L andmoreover, L = Lie G ( A dr ) . [16, 19] The phrase ” diffeographism ” is motivated from this fact that the space Diffg(Φ) acts on cou-pling constants of the theory through formal diffeomorphisms tangent to the identity. Thesediffeographisms together with Birkhoff factorization provide enough information to calculate coun-terterms and renormalized values. Theorem 4.2.5. For the dimensionally regularized Feynman rules character φ ∈ G ( A dr ) , thereis a unique pair ( φ − , φ + ) of characters in G ( A dr ) such that φ = φ − − ∗ φ + . It determines an algebraic Birkhoff decomposition for the chosen character. [17, 24, 37] Based on the Riemann-Hilbert correspondence as a motivation, Connes and Kreimer provedthat physical parameters of a given theory can reformulate with the characters φ − and φ + . It wasthe first bridge between Birkhoff decomposition and theory of quantum fields. Theorem 4.2.6. For the dimensionally regularized Feynman rules character φ in G ( A dr ) ,(i) Its related Birkhoff components are determined by φ − (Γ) = e A dr ◦ ǫ H (Γ) − Υ( φ − ∗ ( φ − e A dr ◦ ǫ H ))(Γ)= − R ms ( φ (Γ) + X γ ⊂ Γ φ − ( γ ) φ ( Γ γ )) ,φ + (Γ) = e A dr ◦ ǫ H (Γ) − e Υ( φ + ∗ ( φ − − e A dr ◦ ǫ H ))(Γ)= φ (Γ) + φ − (Γ) + X γ ⊂ Γ φ − ( γ ) φ ( Γ γ ) , such that Γ ∈ ker ǫ H and the sum is over all disjoint unions of superficially divergent 1PI propersubgraphs.(ii) BPHZ renormalization (i.e. unrenormalized regularized value, counterterms, renormalizedvalues) can be rewritten algebraically by φ (Γ) = U zµ (Γ) , φ − (Γ) = c (Γ) , φ + (Γ) = rv (Γ) . [11, 12, 17, 24, 37] 31o remove divergences step by step in a perturbative expansion of Feynman diagrams is themain idea of renormalization and in this language a theory is renormalizable, if the disappearingof all divergences be possible by such a finite recursive procedure. With attention to the algebraicreformulation of the BPHZ prescription, in the next step of this section, we consider the structureof the renormalization group and its related infinitesimal character. We observe that how geometricconcepts (such as loop space) would be entered in the story to provide a complete algebro-geometricdescription from physical information.Let U zµ (Γ( p , ...p N )) be a regularized unrenormalized value. It determines a loop γ on theinfinitesimal punctured disk ∆ ∗ (connected with the regularization parameter) around the z = 0with values in the group of diffeographisms such that this loop has a Birkhoff factorization γ ( z ) = γ − ( z ) − γ + ( z ) . (4.2.4)where γ − ( z ) (holomorphic in P ( C ) − { } ) gives the counterterm and γ + ( z ) (regular at z = 0)determines the renormalized value. Since U zµ (Γ( p , ...p N )) depends on the mass parameter µ , itsrelated loop should have a dependence on this parameter. In summary, U zµ (Γ( p , ...p N )) ⇐⇒ γ µ : ∆ ∗ −→ Dif f g (Φ) . (4.2.5)Now one can see that the space Loop ( Dif f g (Φ) , µ ) contains physical information of the theoryΦ. Theorem 4.2.7. Let G ( C ) be the pro-unipotent complex Lie group associated to the positivelygraded connected commutative finite type Hopf algebra H of the renormalizable theory Φ underlyingthe minimal subtraction scheme in dimensional regularization. Suppose γ µ ( z ) be a loop with valuesin G ( C ) that encodes U zµ (Γ( p , ..., p n )) . It has a unique Birkhoff decomposition γ µ ( z ) = γ µ − ( z ) − γ µ + ( z ) such that(i) ∂∂µ γ µ − ( z ) = 0 , (ii) For each φ ∈ H ⋆n , t ∈ C : θ t ( φ ) = e nt φ is a − parameter group of automorphisms on G ( C ) , (iii) γ e t µ ( z ) = θ tz ( γ µ ( z )) , (iv) The limit F t = lim z −→ γ − ( z ) θ tz ( γ − ( z ) − ) exists and it denotes a − parameter subgroup F t of G ( C ) . It means that for each s, t , F s + t = F s ∗ F t . (v) For each Feynman diagram Γ , F t (Γ) is a polynomial in t .(vi) ∀ t ∈ R : γ e t µ + (0) = F t γ µ + (0) . [11, 12, 16, 17, 19] Definition 4.2.5. The − parameter subgroup { F t } t of G ( C ) identifies the renormalization groupof the theory such that its infinitesimal generator β = ddt | t =0 F t determines the β − function of the theory. Remark 4.2.6. G ( C ) is a topological group with the topology of pointwise convergence and itmeans that for each sequence { γ n } n of loops with values in G ( C ) and for each Feynman diagram Γ ∈ H , γ n −→ γ ⇐⇒ γ n (Γ) −→ γ (Γ) . g ( C ) be the Lie algebra of diffeographisms of Φ. We know that it contains all of thelinear maps Z : H −→ A satisfying in the Libniz’s law namely, derivations such that its Lie bracketis given by the commutator with respect to the convolution product. There is a bijection betweenthis Lie algebra and its corresponding Lie group given by the exponential map. One can expandthe Lie algebra g ( C ) with an additional generator Z such that for each Z ∈ g ( C ),[ Z , Z ] = Y ( Z ) (4.2.6)where Y is the grading operator. Lemma 4.2.7. (i) dθ t dt | t =0 = Y .(ii) For each φ ∈ H ∗ and Feynman diagram Γ ∈ H , we have < θ t ( φ ) , Γ > = < φ, θ t (Γ) > . It means that θ t = Ad exp ∗ ( tZ ) . [16, 17, 25] In a more practical point of view and for the simplicity in computations, one can have ascattering type formula for components of the dimensionally regularized Feynman rules character φ . It means that Lemma 4.2.8. (i) F t = lim z −→ φ − θ tz φ − − , (ii) β ( φ ) = φ ± ∗ Y ( φ − ± ) = φ ± ∗ [ Z , φ − ± ] , (iii) exp ∗ ( t ( β ( φ ) + Z )) ∗ exp ∗ ( − tZ ) = φ ± ∗ θ t ( φ − ± ) −→ t →∞ φ ± . [25, 26, 87] Finally, let us recalculate physical information to emphasize more the role of the Connes-Kreimer Hopf algebra in this algebraic formalism. Indeed, this machinery works with the antipodemap [61]. If φ be the Feynman rules character, then consider the undeformed character φ ◦ S anddeform this character by the renormalization map. So for each Feynman diagram Γ, the BPHZrenormalization can be summarized in the equations S φR ms (Γ) = − R ms ( φ (Γ)) − R ms ( X γ ⊂ Γ S φR ms ( γ ) φ ( Γ γ )) , (4.2.7)Γ S φR ms ∗ φ (Γ) . (4.2.8)Because it is easy to see that S φR ms ∗ φ (Γ) = R (Γ) + S φR ms (Γ) (4.2.9)such that the Bogoliubov operation R is given by R (Γ) = U zµ (Γ) + X γ ⊂ Γ c ( γ ) U zµ ( Γ γ ) = φ (Γ) + X γ ⊂ Γ S φR ms ( γ ) φ ( Γ γ ) . (4.2.10)It means that S φR ms (Γ) and S φR ms ∗ φ (Γ) are counterterm and renormalized value depended on theFeynman diagram Γ.In summary, Connes-Kreimer perturbative renormalization introduces a new algebraic inter-pretation to calculate the renormalization group based on the loop space of characters and Birkhofffactorization. The geometric nature of this procedure will be shown more, when we consider therenormalization bundle and its related flat equi-singular connections (formulated by Connes andMarcolli). Moreover, we know that the minimal subtraction scheme in dimensional regularizationindicates a Rota-Baxter algebra such that it provides Birkhoff decomposition for characters. Aswe shall see later that how this algebraic property helps us to indicate a new approach to considertheory of integrable systems in renormalizable quantum field theories such that renormalizationhas a central role. 33 hapter 5 Integrable renormalization:Integrable quantum Hamiltoniansystems based on the perturbativerenormalization It was shown that with the help of Hopf algebra of renormalization on Feynman diagrams of arenormalizable QFT and based on components of the Birkhoff decomposition of some particularelements in the loop space on diffeographisms, one can determine counterterms, renormalizedvalues, the Connes-Kreimer renormalization group and its infinitesimal generator. Indeed, theidentification of renormalization with the Riemann-Hilbert problem provides a new conceptualinterpretation of physical information.With applying Atkinson theorem, one can show that the existence and the uniqueness of thisfactorization is a direct consequence of an important and interesting concept in physics namely,multiplicativity of renormalization which is prescribed mathematically with the Rota-Baxter con-dition of the chosen renormalization scheme. On the other hand and in a roughly speaking, one canfind the importance of this class of equations in the study of theory of integrable systems specially,(modified) Yang-Baxter equations. This fact reports obviously a pure algebraic configuration ofrenormalization where it indeed seems wise to introduce a coherent ideology for considering quan-tum integrable systems with respect to the Connes-Kreimer theory. [23, 25, 26, 59, 65]With attention to the theory of integrable systems in classical level [8, 20, 63, 64, 65, 86, 89],this (rigorous) algebraic machinery for the description of renormalizable QFTs and the power ofnoncommutative differential forms [35], we are going to find a new family of Hamiltonian systemswhich arise from the Connes-Kreimer approach to perturbative renormalization and moreover,we show that how integrability condition can be determined naturally based on Poisson bracketsrelated to Rota-Baxter type algebras. The beauty of this new viewpoint to integrable systemswill be identified, when we consider the minimal subtraction underlying the dimensional regular-ization (as the renormalization mechanism) and then study possible relation between introducedmotion integrals and renormalization group. After that, based on the Rosenberg framework, wefamiliar with other group of quantum Poisson brackets and their related motion integrals. Weclose this chapter with introducing a new family of fixed point equations modified with motionintegral condition and Bogoliubov character. In summary, it seems favorable to report about theappearance of a glimpse of one general relation between the theory of Rota-Baxter type algebrasand the Riemann-Hilbert problem underlying quantum field theory. [92, 94, 95]34 .1 What is an integrable system? From finite dimension (geometricapproach) to infinite dimension (algebraic approach) Let V be a m -dimensional real vector space and ω : V × V −→ R a skew-symmetric bilinear map. Definition 5.1.1. For a map e ω : V −→ V ∗ given by e ω ( v )( u ) := ω ( v, u ) ,ω is called a symplectic form, if e ω be a bijective map. The pair ( V, ω ) is called symplectic vectorspace and it is clear that each symplectic vector space has even dimension. Definition 5.1.2. A differential 2-form ω on a manifold M is called symplectic, if it is closedand for each p ∈ M , ω p is a symplectic form on T p M . Definition 5.1.3. (i) A complex structure on V is a linear map J : V −→ V such that J = − Id V . ( V, J ) is called a complex vector space.(ii) An almost complex structure on a manifold M is a smooth field of complex structures onthe tangent spaces and it means that for each x ∈ M , x J x : T x M −→ T x M : Linear, J x = − Id. The pair ( M, J ) is called an almost complex manifold.(iii) An almost complex structure J is called integrable, if J is induced by a structure of complexmanifold on M . There is a well known operator to characterize integrable almost complex structures on mani-folds. It will be shown that this map is in fact a starting key for us to consider quantum integrablesystems underlying an algebraic formalism. Definition 5.1.4. For the almost complex manifold ( M, J ) , the Nijenhuis tensor is defined by N T ( v, w ) := [ Jv, Jw ] − J [ v, Jw ] − J [ Jv, w ] − [ v, w ] such that v, w are vector fields on M and for each f ∈ C ∞ ( M ) , [ v, w ] .f = v. ( w.f ) − w. ( v.f ) where v.f = df ( v ) . Theorem 5.1.1. For the almost complex manifold ( M, J ) , the following facts are equivalent:(i) M is a complex manifold,(ii) J is integrable,(iii) N T ≡ ,(iv) d = ∂ + ∂ . [3, 8] Definition 5.1.4 and theorem 5.1.1 show that if J is an integrable structure, then for each x ∈ M and v, w ∈ T x M , we have[ J x v, J x w ] = J x [ v, J x w ] + J x [ J x v, w ] + [ v, w ] . (5.1.1)Equations of motion are the results of variational problems in classical mechanics and in asystem with n particles in R n , all of the physical trajectories follow from the Newton’s secondlaw. In these paths the mean value of the difference between kinetics and potential energies isminimum. 35 emma 5.1.5. Let ( M, ω ) be a symplectic manifold and H : M −→ R a smooth function. Thereis a unique vector field X H on M such that i X H ω = dH . One can identify a 1-parameter familyof diffeomorphisms ρ t : M −→ M such that ρ = Id M , dρ t dt ◦ ρ − t = X H , ρ ∗ t ω = ω. [3, 8, 63] This lemma shows that each smooth function H : M −→ R determines a family of symplec-tomorphisms . The function H is called Hamiltonian function such that the vector field X H is itscorresponding Hamiltonian vector field . Definition 5.1.6. The triple ( M, ω, H ) is called a classical Hamiltonian system. On the Euclidean space R n with the local coordinate ( q , ..., q n , p , ..., p n ) and the canonicalsymplectic form ω = P dq i ∧ dp i , the curve α t = ( q ( t ) , p ( t )) is an integral curve of the vector field X H if and only if it determines the Hamiltonian equations of motion dq i dt = ∂H∂p i , dp i dt = − ∂H∂q i (5.1.2)on R n . Theorem 5.1.2. The Newton’s second law on the configuration space R n is equivalent to theHamiltonian equations of motion on the phase space R n . [3, 8, 63] For a given symplectic manifold M and f, g ∈ C ∞ ( M ), letting X f , X g be the Hamiltonianvector fields with respect to these functions. From this class of vector fields, one expects a Poissonbracket on C ∞ ( M ) given by { f, g } := ω ( X f , X g ) . (5.1.3)The pair ( C ∞ ( M ) , { ., . } ) is called Poisson algebra related to the configuration space M and fromnow M is called a Poisson manifold . Remark 5.1.7. Dual space of a Lie algebra is one useful example of a Poisson manifold butgenerally, these Poisson manifolds are not symplectic. Lemma 5.1.8. For any H ∈ C ∞ ( M ) , its related Hamiltonian vector field which acts on theelements of C ∞ ( M ) (by the Poisson bracket) can be rewritten with the equation X H f = { H, f } such that for each x ∈ M , the vectors X H ( x ) span a linear subspace in T x M . [3, 8, 63] We know that Hamiltonian vector fields are tangent to symplectic leaves and it means thatthe Hamiltonian flows enable to preserve each leaf separately. So for a given Hamiltonian system( M, ω, H ), equation { f, H } = 0 (5.1.4)is equivalent with this fact that f is constant along the integral curves of X H . The function f iscalled integral of motion . [8, 86] Definition 5.1.9. The Hamiltonian system ( M, ω, H ) is called integrable, if there exist n = dimM independent integrals of motion f = H, f , ..., f n such that { f i , f j } = 0 . [3, 8] g be a Lie algebra with the dual space g ⋆ and P ( g ⋆ ) be the space of polynomials onthe dual space. Elements of the Lie algebra can determine a bracket on P ( g ⋆ ) such that for each g , g ∈ g and h ⋆ ∈ g ⋆ , it is given by { g , g } ( h ⋆ ) := h ⋆ ([ g , g ]) . (5.1.5)Since P ( g ⋆ ) is dense in C ∞ ( g ⋆ ), one can expand canonically this bracket to obtain a Lie Poissonbracket such that for each s , s ∈ C ∞ ( g ⋆ ), we have { s , s } ( h ⋆ ) = h ⋆ ([ ds ( h ⋆ ) , ds ( h ⋆ )]) . (5.1.6) Theorem 5.1.3. For a given Lie group G with the Lie algebra g , the symplectic leaves of (5.1.6)are G − orbits in g ⋆ . [3, 63, 89] One can apply automorphisms on a Lie algebra to introduce other Lie Poisson brackets. Thismethod provides a new favorable point of view to study infinite dimensional integrable systems. Definition 5.1.10. For a fixed Lie algebra g , an endomorphism R ∈ End ( g ) is called R-matrix,if for each g , g ∈ g , the bracket [ g , g ] R = 12 ([ R ( g ) , g ] + [ g , R ( g )]) satisfies in the Jacobi identity and it means that [ ., . ] R is a Lie bracket. Yang-Baxter equation is enough to introduce a new class of Poisson brackets on g ⋆ which arisefrom this kind of Lie brackets. Theorem 5.1.4. Let R ∈ End ( g ) . For each g , g ∈ g , set B R ( g , g ) := [ R ( g ) , R ( g )] − R ([ R ( g ) , g ] + [ g , R ( g )]) . The R − bracket defined in 5.1.10 follows the Jacobi identity if and only if for each g , g , g ∈ g ,we have [ B R ( g , g ) , g ] + [ B R ( g , g ) , g ] + [ B R ( g , g ) , g ] = 0 . This condition is called classical Yang-Baxter equation. The simplest sufficient condition is givenby B R ( g , g ) = 0 . [63, 64, 65, 89] Yang-Baxter equations make possible to consider integrable systems in an algebraic frameworksuch that this ability together with the Hopf algebra of Feynman diagrams help us to study quan-tum Hamiltonian systems and integrability condition depended on the algebraic renormalization.The critical point in this procedure can be summarized in reinterpretation of the renormaliza-tion group on the basis of Rota-Baxter type algebras. We show that how one can find a newfamily of symplectic structures where its existence is strongly connected with the choosing ofrenormalization prescription. Rota-Baxter type algebras: Nijenhuis algebras In this part we dwell a moment on a special class of Rota-Baxter type algebras namely, Nijenhuisalgebras to discuss that how one can apply these maps to deform the initial product. It gives afamily of new associative algebras together with related compatible Lie brackets.For a linear map N : A −→ A , define a new product on A . It is given by( x, y ) x ◦ N y := N ( x ) y + xN ( y ) − N ( xy ) . (5.2.1)37 emark 5.2.1. If e be the unit of ( A, m ) and N ( e ) = e then ( A, ◦ N ) has the same unit. Lemma 5.2.2. (i) ( A, ◦ N ) is an algebra.(ii) The product ◦ N is associative if and only if for each x, y ∈ A , T N ( x, y ) := N ( x ◦ N y ) − N ( x ) N ( y ) be a Hochschild 2-cocycle of the algebra A (with respect to the Hochschild coboundary operator b connected with the product of A ). It means that for each x, y, z ∈ A , b T N ( x, y, z ) := xT N ( y, z ) − T N ( xy, z ) + T N ( x, yz ) − T N ( x, y ) z = 0 . (iii) N is a derivation in the original algebra if and only if N is a 1-cocycle with respect to thecoboundary operator b . In this case, the new product ◦ N is trivial. [9, 21] Definition 5.2.3. The linear map N is called Nijenhuis tensor supported by ◦ N , if N ( x ◦ N y ) = N ( x ) N ( y ) . (5.2.2) The pair ( A, N ) is called Nijenhuis algebra. Remark 5.2.4. (i) If N is a Nijenhuis tensor, then ◦ N is an associative product on A and alsofor each λ ∈ K , m + λ ◦ N is an associative product on A (such that m is the original product of A ).(ii) For a given Rota-Baxter algebra ( A, R ) with the idempotent Rota-Baxter map R and foreach λ ∈ K , the operator N λ := R − λ e R has Nijenhuis property and it means that ( A, N λ ) is aNijenhuis algebra. [24, 25, 26] The classical Yang-Baxter equation is essentially governed by the extension of this kind ofoperators to the Lie algebra level. Definition 5.2.5. A Nijenhuis tensor for the Lie algebra ( A, [ ., . ]) is a linear map N : A −→ A such that for each x, y ∈ A , N ([ x, y ] N ) = [ N ( x ) , N ( y )] where [ x, y ] N := [ N ( x ) , y ] + [ x, N ( y )] − N ([ x, y ]) . Remark 5.2.6. The compatibility of this Lie bracket is strongly related to the Nijenhuis propertyof N and it is easy to see that in this case [ N ( x ) , N ( y )] = N ([ N ( x ) , y ]) + N ([ x, N ( y )]) − N ([ x, y ]) . The triple ( A, [ ., . ] , N ) is called Nijenhuis Lie algebra. [21] Theorem 5.2.1. Let ( A, ◦ N ) be the associative algebra with respect to the Nijenhuis tensor N on A . N is a Nijenhuis tensor for the Lie algebra ( A, [ ., . ]) (i.e. [ ., . ] is the commutator with respectto the product m ) and for each x, y ∈ A , we will have [ x, y ] N = x ◦ N y − y ◦ N x. (5.2.3) [9, 21, 24, 25, 26] Because of the importance of Nijenhuis algebras, we are going to mention an algorithmicinstruction for constructing this type of algebras from each arbitrary commutative algebra. Let( A, m ) be a commutative K − algebra with the tensor algebra T ( A ) := ⊕ n ≥ A ⊗ n . Elements of A play the role of letters and generators U = a ⊗ ... ⊗ a n in A ⊗ n (where a i ∈ A ) are identified with words a ...a n . 38 emma 5.2.7. For each a, b ∈ A, U ∈ A ⊗ n , V ∈ A ⊗ m and λ ∈ K , one can define recursively anassociative commutative quasi-shuffle product on T ( A ) given by aU ⋆ bV := a ( U ⋆ bV ) + b ( aU ⋆ V ) − λm ( a, b )( U ⋆ V ) such that the empty word plays the role of its unit. [22, 23, 39] The product given by the lemma 5.2.7 determines another kind of shuffle product. Lemma 5.2.8. An augmented quasi-shuffle product on the augmented tensor module T ( A ) := ⊕ n> A ⊗ n is given by aU ⊙ bV := m ( a, b )( U ⋆ V ) . It is an associative commutative product such that the unit e of the algebra A is the unit for thisproduct. [22, 23, 39] One can define a modified version of the above products. Lemma 5.2.9. The modified quasi-shuffle product on T ( A ) is defined by aU ⊖ bV := a ( U ⊖ bV ) + b ( aU ⊖ V ) − em ( a, b )( U ⊖ V ) such that the empty word is the unit for this commutative associative product. Its augmentedversion namely, augmented modified quasi-shuffle product on T ( A ) is defined by aU ⊘ bV := m ( a, b )( U ⊖ V ) such that the unit e of A is the unit for this new product. [22, 23, 39] These products can be applied to make a special family of Rota-Baxter and Nijenhuis algebraswith the universal property. Theorem 5.2.2. For a unital commutative associative K − algebra A with the unit e , let B + e be alinear operator on T ( A ) given by B + e ( a ...a n ) := ea ...a n . (i) ( T ( A ) , ⋆, λ, B + e ) is a RB algebra of weight λ .(ii) ( T ( A ) , ⊖ , B + e ) is a Nijenhuis algebra.(iii) ( T ( A ) , ⊙ , λ, B + e ) is the universal RB algebra of weight λ generated by A .(iv) ( T ( A ) , ⊘ , B + e ) is the universal Nijenhuis algebra generated by A . [22] Remark 5.2.10. For instance, universal Nijenhuis algebra (generated by A ) means that for anyNijenhuis algebra B and algebra homomorphism f : A −→ B , there exists a unique Nijenhuishomomorphism e f : T ( A ) −→ B such that e f ◦ j A = f . Everything is prepared to introduce a new class of quantum Hamiltonian systems as the conse-quence of the Connes-Kreimer algebraic framework to perturbative theory. It will be shown thathow the Hopf algebraic renormalization group can give us some examples of integrable systemsfrom this class of Hamiltonian systems. 39 .3 Theory of quantum integrable systems In this part we want to focus on the algebraic basis of the Connes-Kreimer theory namely, theRota-Baxter property induced from renormalization to improve theory of integrable systems to thelevel of renormalizable physical theories. With the help of noncommutative differential forms (as-sociated with Hamiltonian derivations) and with attention to the chosen renormalization method(i.e. regularization algebra and renormalization map), we are going to introduce a new familyof Hamiltonian systems based on the Connes-Kreimer Hopf algebra of Feynman diagrams. Itis discussed that how integrability condition on these systems are strongly connected with theperturbative renormalization process. [94, 95]Renormalization prescription makes possible two different types of deformations. In one class,we consider deformed algebras which deformation process is performed by an idempotent renor-malization map and in another class, with respect to regularization scheme (independent of therenormalization map), we turn to Ebrahimi-Fard’s aspect in defining the universal Nijenhuis al-gebra (as the kind of deformation method) and then we will deform the initial algebra by thisuniversal Nijenhuis tensor. Finally, with working on differential forms of these deformed algebras,we will illustrate (as the conclusion) Hamiltonian systems and also integrability condition.Roughly speaking, Connes-Kreimer Hopf algebra and regularization algebra determines a newnoncommutative algebra such that with attention to the renormalization scheme, one can pro-vide new deformed algebras. Then with working on the noncommutative differential calculuswith respect to these mentioned deformed algebras, we can obtain symplectic structures and soHamiltonian systems. Because of applying this noncommutative differential formalism on the de-formed algebras depended upon the renormalization process, it does make sense to use the phrase” integrable renormalization ” for this new captured approach to quantum integrable systems. First Class Suppose Φ be a renormalizable QFT with the associated Hopf algebra of Feynman diagrams H and letting the perturbative renormalization is performed with the idempotent renormalizationmap R and the regularization scheme A . In summary, we denote its algebraic reformulation with e Φ = ( L ( H, A ) , ∗ , Υ) (5.3.1)such that the idempotent Rota-Baxter map Υ on L ( H, A ) is given by R . It is easy to show thatfor each λ ∈ K , the operator Υ λ := Υ − λ e Υ (where e Υ := Id − Υ) has Nijenhuis property. Definition 5.3.1. By the formula (5.2.1), for each λ a new product ◦ λ on L ( H, A ) can be definedsuch that for each φ , φ ∈ L ( H, A ) , φ ◦ λ φ := Υ λ ( φ ) ∗ φ + φ ∗ Υ λ ( φ ) − Υ λ ( φ ∗ φ ) . Remark 5.3.2. (i) One can show that Υ λ ( φ ◦ λ φ ) = Υ λ ( φ ) ∗ Υ λ ( φ ) . (ii) Lemma 5.2.2 shows that ◦ λ is an associative product and one can identify the followingcompatible Lie bracket [ φ , φ ] λ := [Υ λ ( φ ) , φ ] + [ φ , Υ λ ( φ )] − Υ λ ([ φ , φ ]) such that definition 5.3.1 provides that [ φ , φ ] λ = φ ◦ λ φ − φ ◦ λ φ . It means that one can extend the Nijenhuis property of Υ λ to the Lie algebra level (with thecommutator with respect to the product ∗ ). efinition 5.3.3. The new spectral information ( e Φ λ , [ ., . ] λ ) := ( L ( H, A ) , ◦ λ , Υ λ , [ ., . ] λ ) is called λ − information based on the theory Φ and with respect to the Nijenhuis map Υ λ . Second Class Generally, when we study the algebraic renormalization methods, the renormalization mapmight not have idempotent property and therefore we should apply another technique to receiveNijenhuis tensors. In this situation one can focus on a commutative algebra (which reflects theregularization scheme) and apply the universal Nijenhuis tensor based on this algebra.Let Φ be a renormalizable theory with the related Hopf algebra H such that the regular-ization scheme is given by the associative commutative unital algebra A . Theorem 5.2.2 showsthat ( T ( A ) , ⊘ , B + e ) is the universal Nijenhuis algebra based on A and therefore one can define aNijenhuis map Υ + e on L ( H, T ( A )) given byΥ + e ( ψ ) := B + e ◦ ψ. (5.3.2) Definition 5.3.4. With help of the operator Υ + e , a new product ◦ u on L ( H, T ( A )) is introducedsuch that for each ψ , ψ ∈ L ( H, T ( A )) , ψ ◦ u ψ := Υ + e ( ψ ) ∗ ⊘ ψ + ψ ∗ ⊘ Υ + e ( ψ ) − Υ + e ( ψ ∗ ⊘ ψ ) , where ψ ∗ ⊘ ψ := ⊘ ( ψ ⊗ ψ ) ◦ ∆ H . Remark 5.3.5. (i) The Nijenhuis property of Υ + e provides that Υ + e ( ψ ◦ u ψ ) = Υ + e ( ψ ) ∗ ⊘ Υ + e ( ψ ) , such that it supports the associativity of the product ◦ u .(ii) A compatible Lie bracket [ ., . ] u can be defined on L ( H, T ( A )) by [ ψ , ψ ] u := [Υ + e ( ψ ) , ψ ] + [ ψ , Υ + e ( ψ )] − Υ + e ([ ψ , ψ ]) . Moreover we have [ ψ , ψ ] u = ψ ◦ u ψ − ψ ◦ u ψ . It means that Nijenhuis property of Υ + e can be extended to the Lie algebra level (with the commu-tator with respect to the product ∗ ⊘ ). Definition 5.3.6. The spectral information ( e Φ u , [ ., . ] u ) := ( L ( H, T ( A )) , ◦ u , Υ + e , [ ., . ] u ) is called u − information based on the theory Φ and with respect to the Nijenhuis map Υ + e . Symplectic Structures Here we introduce a certain family of symplectic structures related to this algebraic prepara-tion from renormalizable physical theories and for this goal we need theory of noncommutativedifferential calculus over an algebra based on the space of its derivations. Definition 5.3.7. Let C be an associative unital algebra over the field K (with characteristic zero)with the center Z ( C ) . A derivation θ : C −→ C is an infinitesimal automorphism of C such thatit is a linear map satisfying the Leibniz rule where if it could have been exponentiated, then themap exp θ will be an automorphism of C . emark 5.3.8. (i) Geometrically, θ is a vector field on a noncommutative space and t exp( tθ ) is the one parameter flow of automorphisms (i.e. integral curves) of our noncommutative spacegenerated by θ .(ii) Suppose Der ( C ) be the space of all derivations on C . It is a module over Z ( C ) such thatit has a Lie algebra structure with the Lie bracket given by the commutator with respect to thecomposition of derivations. [35, 102, 103] Definition 5.3.9. Letting Ω nDer ( C ) be the space of all Z ( C ) − multilinear antisymmetric map-pings from Der ( C ) n into C such that Ω Der ( C ) = C . A differential graded algebra Ω • Der ( C ) = L n ≥ Ω nDer ( C ) can be defined where for each ω ∈ Ω nDer ( C ) and θ i ∈ Der ( C ) , its antiderivationdifferential operator d of degree one is given by ( dω )( θ , ..., θ n ) := n X k =0 ( − k θ k ω ( θ , ..., b θ k , ..., θ n ) + X ≤ r The noncommutative deRham complex on Der ( C ) is defined by DR • Der ( C ) := Ω • Der ( C )[Ω • Der ( C ) , Ω • Der ( C )] . We know that an algebra C equipped with a bi-derivation Lie bracket { ., . } which satisfies inthe Jacobi identity determines a Poisson algebra . There is a class of derivations on this algebrawhich contains an essential geometric meaning. Definition 5.3.12. For each c in the Poisson algebra C , derivation ham ( c ) : x c, x } is called a Hamiltonian derivation (vector field) corresponding to c . Definition 5.3.13. For a given Poisson algebra C , a Z ( C ) − bilinear antisymmetric map ω in Ω Der ( C ) is called non-degenerate, if for any element c ∈ C , there exists a derivation θ c of C suchthat for each derivation θ , ω ( θ c , θ ) = θ ( c ) . In this case derivation θ c = ham ( c ) is unique and the function θ i θ ω is linear and injectiveand it is observed that ( i ham ( c ) ω )( θ ) = ω ( ham ( c ) , θ ) = θ ( c ) =: ( dc )( θ ) such that dc : Der ( C ) −→ C is a 1-form. efinition 5.3.14. A closed non-degenerate element ω in Ω Der ( C ) is called a symplectic struc-ture. With the help of this symplectic form, one can define an antisymmetric bilinear bracket on C such that for each x, y ∈ C , it is given by { x, y } ω := ω ( ham ( x ) , ham ( y )) . (5.3.3)It satisfies Leibniz law and Jacobi identity and therefore it determines a Poisson bracket on C . Lemma 5.3.15. (i) There is a Lie algebra homomorphism from ( C, { ., . } ω ) to ( Der ( C ) , [ ., . ]) .(ii) When Z ( C ) − module generated by the set Ham ( C ) := { ham ( c ) : c ∈ C } be the entire of the space Der ( C ) , the Jacobi identity for the bracket { ., . } ω and closed conditionfor the symplectic structure ω are equivalent. [35, 36, 102, 103]Proof. One can show that [ ham ( x ) , ham ( y )] = ham ( { x, y } ω ) . It means that ham plays the role of a Lie homomorphism.If a given associative algebra C follows condition (ii) in the lemma 5.3.15, then the Poissonbracket on C is called non-degenerate and otherwise it is called degenerate . It is possible to reachto a symplectic structure from a non-degenerate Poisson bracket. Lemma 5.3.16. For a non-degenerate Poisson bracket { ., . } on the associative algebra C , thereexists a symplectic structure ω such that its related Poisson bracket coincides with { ., . } . [35, 36]Proof. Let θ , θ be derivations on C . From non-degeneracy of the Poisson bracket, we know that Z ( C ) .Ham ( C ) = Der ( C ) . Therefore there exist { x , ..., x m , y , ..., y n } ⊂ C and { u , ..., u m , v , ..., v n } ⊂ Z ( C ) such that θ = X i u i ham ( x i ) , θ = X j v j ham ( y j ) . Define ω ( θ , θ ) := X i,j u i v j { x i , y j } .ω is our interesting symplectic structure.Now we are ready to combine the theory of noncommutative differential forms with the Connes-Kreiemr approach to obtain a new family of Hamiltonian systems completely depended upon therenormalization procedure. At first suppose in theory Φ one can perform renormalization with anidempotent Rota-Baxter renormalization map R and the related λ − information e Φ λ ( λ ∈ K ). TheLie bracket [ ., . ] λ determines a Poisson bracket in a natural way where lemma 5.3.16 can inducea symplectic structure ω λ but for this result we need the non-degeneracy of this Poisson bracketsuch that in general maybe it does not happen. According to the proof of this lemma and since forthe identification of integrals of motion just we should concentrate on Hamiltonian vector fields(derivations), therefore for removal this problem it is enough to work on- Z ( e Φ λ ) − module Der Ham ( e Φ λ ) generated by the set Ham ( e Φ λ ) (i.e. all Hamiltonian derivationsof the algebra e Φ λ ) instead of the set of all derivations Der ( e Φ λ ).- And restrict the differential graded algebra Ω • Der ( e Φ λ ) into the differential graded algebraΩ • Der Ham ( e Φ λ ) such that Ω Der Ham ( e Φ λ ) = e Φ λ and Ω nDer Ham ( e Φ λ ) is the space of all Z ( e Φ λ ) − multilinearantisymmetric mappings from Der Ham ( e Φ λ ) n into e Φ λ .43 orollary 5.3.17. Symplectic forms related to the first class . The differential form ω λ : Der Ham ( e Φ λ ) × Der Ham ( e Φ λ ) −→ e Φ λ in Ω Der Ham ( e Φ λ ) given by ω λ ( θ, θ ′ ) := X i,j u i ◦ λ v j ◦ λ [ f i , h j ] λ such that { f , ..., f m , h , ..., h n } ⊂ L ( H, A ) , { u , ..., u m , v , ..., v n } ⊂ Z ( e Φ λ ) and θ = P i u i ◦ λ ham ( f i ) , θ ′ = P j v j ◦ λ ham ( h j ) is a Z ( e Φ λ ) − bilinear, antisymmetric, non-degenerate and closedelement (i.e. a symplectic structure) . At second if renormalization map does not have idempotent property or in general, then onecan concentrate on the universal Nijenhuis tensor to obtain a Poisson bracket [ ., . ] u . Maybe it hasnot non-degeneracy and so for determining a symplectic structure ω u , it is enough to focus on- Z ( e Φ u ) − module Der Ham ( e Φ u ) generated by the set Ham ( e Φ u ).- And restrict the differential graded algebra Ω • Der ( e Φ u ) into the differential graded algebraΩ • Der Ham ( e Φ u ) such that Ω Der Ham ( e Φ u ) = e Φ u and Ω nDer Ham ( e Φ u ) is the space of all Z ( e Φ u ) − multilinearantisymmetric mappings from Der Ham ( e Φ u ) n into e Φ u . Corollary 5.3.18. Symplectic forms related to the second class . The differential form ω u : Der Ham ( e Φ u ) × Der Ham ( e Φ u ) −→ e Φ u in Ω Der Ham ( e Φ u ) given by ω u ( ρ, µ ) := X i,j w i ◦ u z j ◦ u [ g i , k j ] u such that { g , ..., g m , k , ..., k n } ⊂ L ( H, T ( A )) , { w , ..., w m , z , ..., z n } ⊂ Z ( e Φ u ) and ρ = P i w i ◦ u ham ( g i ) , µ = P j z j ◦ u ham ( k j ) is a symplectic structure. Proposition 5.3.19. The Connes-Kreimer algebraic perturbative renormalization (namely, thecouple renormalization map and regularization algebra) determines a family of symplectic struc-tures. Now it is convenient to use the terminology ” symplectic space ” for this new information ob-tained form the geometric studying of renormalization. Definition 5.3.20. Symplectic spaces related to the Hopf algebraic reconstruction of the pertur-bative renormalization in a renormalizable QFT are introduced by Γ λ := ( e Φ λ , Der Ham ( e Φ λ ) , ω λ ) , and Γ u := ( e Φ u , Der Ham ( e Φ u ) , ω u ) . Lemma 5.3.21. Suppose θ be a derivation in Der Ham ( e Φ λ ) and ρ be a derivation in Der Ham ( e Φ u ) .One can show that L θ ω λ = i θ ◦ d λ ω λ + d λ ◦ i θ ω λ = d λ ( i θ ω λ ) = 0 and L ρ ω u = i ρ ◦ d u ω u + d u ◦ i ρ ω u = d u ( i ρ ω u ) = 0 such that d λ ( d u ) is the anti-derivation differential operator for the differential graded algebra Ω • Der Ham ( e Φ λ ) ( Ω • Der Ham ( e Φ u ) ). This derivation is called λ − symplectic ( u − symplectic) vector fieldwith respect to the symplectic structure ω λ ( ω u ). orollary 5.3.22. Maps i ω λ : Der Ham ( e Φ λ ) −→ DR Der Ham ( e Φ λ ) ,θ i θ ω λ and i ω u : Der Ham ( e Φ u ) −→ DR Der Ham ( e Φ u ) ,ρ i ρ ω u are well defined, linear and bijection. It means that one can find a bijection between closed oneforms in DR Der Ham ( e Φ λ ) (or DR Der Ham ( e Φ u ) ) and λ − symplectic (or u − symplectic) vector fields.Proof. For each λ ∈ K , it is enough to know that ω λ ( ω u ) is non-degenerate and closed. Definition 5.3.23. If θ λf be the λ − symplectic vector field associated with d λ f such that f ∈ Ω Der Ham ( e Φ λ ) = e Φ λ . Then a new Poisson bracket on e Φ λ is defined by { f, g } λ := i θ λf ( d λ g ) . It is called λ − Poisson bracket. Corollary 5.3.24. (i) For each λ ∈ K , there is a new Lie algebra structure on e Φ λ such that themap ( e Φ λ , { ., . } λ ) −→ ( Der Ham ( e Φ λ ) , [ ., . ]) is a Lie algebra homomorphism.(ii) λ − Poisson bracket { ., . } λ is characterized with the symplectic structure ω λ (and thereforeby the Lie brackets [ ., . ] λ ). It means that { f, g } λ = i θ λf ( d λ g ) = i θ λf i θ λg ω λ , i [ θ λf ,θ λg ] ω λ = d λ i θ λf ( d λ g ) = d λ { f, g } λ . Corollary 5.3.25. Derivation θ λ { f,g } λ is the unique λ − symplectic vector field with respect to d λ { f, g } λ and it means that θ λ { f,g } λ = [ θ λf , θ λg ] . There is a similar process for the symplectic space Γ u and one can obtain a u − Poisson bracket { ., . } u on e Φ u induced with the symplectic structure ω u (and therefore by the Lie bracket [ ., . ] u ). Remark 5.3.26. If the Poisson bracket [ ., . ] λ (or [ ., . ] u ) is non-degenerate, then we can definesymplectic structure ω λ (or ω u ) on the whole space of derivations of e Φ λ (or e Φ u ) and thereforelemma 5.3.16 shows that the Poisson bracket { ., . } λ (or { ., . } u ) will be coincide with [ ., . ] λ (or [ ., . ] u ). But in general, might be they are not the same. Hamiltonian Systems In classical mechanics a Hamiltonian system consists of a symplectic manifold (as a configu-ration space) and a Hamiltonian operator such that a Poisson bracket can be inherited from thisinformation. Then naturally, related motion integrals would be determined while as the resultwe will enable to identify integrable systems. Here we want to carry out the same procedure toobtain a new family of infinite Hamiltonian systems depended upon renormalizable perturbativetheories. Definition 5.3.27. (i) For each λ ∈ K and a fixed F ∈ L ( H, A ) , the pair (Γ λ , F ) is called λ − Hamiltonian system.(ii) For G ∈ L ( H, T ( A )) , the pair (Γ u , G ) is called u − Hamiltonian system with respect to thetheory Φ and the regularization scheme A .(iii) F and G are called Hamiltonian functions. Der ( e Φ λ ) (or Der ( e Φ u )) given by thecomposition of derivations, we could characterize integrals of motion. Proposition 5.3.28. (i) Let Φ be a renormalizable theory with the idempotent renormalizationmap R and (Γ λ , F ) be a λ − Hamiltonian system with respect to it. { f, F } λ = 0 ⇐⇒ The map f is constant along the integral curves of θ λF (i.e. 1-parameter flow α λt : t exp ( tθ λF ) of the automorphisms generated by θ λF ).The function f is called λ − integral of motion of this system.(ii) Let Φ be a renormalizable theory with the regularization scheme A and (Γ u , G ) be a u − Hamiltonian system with respect to it. { g, G } u = 0 ⇐⇒ The map g is constant along the integral curves of ρ uG (i.e. 1-parameter flow α ut : t exp ( tρ uG ) of the automorphisms generated by ρ uG ).The function g is called u − integral of motion for this system.Proof. With using the Cartan magic formula [8], we have ddt ( α λt ) ∗ ( f ) = ( α λt ) ∗ L θ λF f = ( α λt ) ∗ i θ λF d λ f = ( α λt ) ∗ i θ λF i θ λf ω λ = ( α λt ) ∗ ω λ ( θ λf , θ λF ) = ( α λt ) ∗ { f, F } λ = 0 . And finally, the concept of integrability (based on the behavior of motion integrals) for thisclass of quantum Hamiltonian systems can be determined in an usual procedure. It means that Definition 5.3.29. (i) A λ − Hamiltonian system (Γ λ , F ) is called ( n, λ ) − integrable, if there exist n linearly independent λ − integrals of motion f = F, f , ..., f n such that { f i , f j } λ = 0 .(ii) A u − Hamiltonian system (Γ u , G ) is called ( n, u ) − integrable, if there exist n linearly inde-pendent u − integrals of motion g = G, g , ..., g n such that { g i , g j } u = 0 .(iii) If a λ − Hamiltonian (or u − Hamiltonian) system has infinite linearly independent λ − (or u − )integrals of motion, then it is called infinite dimensional λ − integrable ( or infinite dimensional u − integrable) system. Briefly speaking, we could provide the concept of integrability of Hamiltonian systems in renor-malizable QFTs with respect to regularization algebras or idempotent renormalization schemessuch that integrals of motion are introduced in consistency of determined Nijenhuis operators.This process enables to reflect the dependency of this style of (integrable) Hamiltonian systemsupon the perturbative renormalization. More precisely, there is an interesting chance to checkthe strong compatibility of this approach with the Connes-Kreimer formalism and it can be doneby the renormalization group. Actually, we will show that using Connes-Kreimer renormalizationgroup can guide us to characterize some examples of integrable quantum Hamiltonian systems. Infinite integrable quantum Hamiltonian systems on the basis ofthe renormalization group We saw that how one can reach to a fundamental concept of Hamiltonian formalism from defor-mation of Connes-Kreimer convolution algebras. It is the place to show further the compatibility46f this approach with the Hopf algebraic machinery in terms of the renormalization group. So inthis part, we work on the BPHZ prescription to recognize motion integrals depended on charactersof the renormalization Hopf algebra and also, we apply the Connes-Kreimer Birkhoff factorizationto determine some conditions for components of this factorization on a Feynman rules character φ when they are motion integrals of φ .Roughly, here we consider the relation between defined Nijenhuis type motion integrals andphysical information underlying minimal subtraction in dimensional regularization [95]. Particu-larly, we further concentrate on Hamiltonian systems such that their Hamiltonian functions aredimensionally regularized Feynman rules characters or elements of the Connes-Kreimer renormal-ization group.Starting with a renormalizable physical theory Φ together with the dimensionally regularizedFeynman rules character φ underlying the renormalization prescription ( A dr , R ms ). We knowthat R ms is a Nijenhuis tensor and so for each λ ∈ K , the map Υ ms,λ on L ( H, A dr ) has alsothis property such that one can extend it to the Lie algebra level. Since we need to study thebehavior of the renormalization group, so just we focus on the case λ = 0 and note that thereare similar calculations for other values of λ . Consider the Rota-Baxter map Υ ms where for each φ ∈ L ( H, A dr ), Υ ms ( φ ) := R ms ◦ φ. (5.4.1)Induce a new associative product and a Lie bracket on L ( H, A dr ) given by φ ◦ φ := R ms ◦ φ ∗ φ + φ ∗ R ms ◦ φ − R ms ◦ ( φ ∗ φ ) . (5.4.2)[ φ , φ ] := [ R ms ◦ φ , φ ] + [ φ , R ms ◦ φ ] − R ms ◦ ([ φ , φ ]) . (5.4.3)Equation (5.2.5) shows that R ms ◦ [ φ , φ ] = [ R ms ◦ φ , R ms ◦ φ ] . (5.4.4) Definition 5.4.1. The − information based on the theory Φ and with respect to the map Υ ms isdefined by ( e Φ , [ ., . ] ) := ( L ( H, A dr ) , ◦ , Υ ms , [ ., . ] ) . Corollary 5.3.17 determines a symplectic structure for this 0 − information. For arbitrary deriva-tions θ = P i u i ◦ ham ( f i ) and θ ′ = P j v j ◦ ham ( h j ) in Der Ham ( e Φ ) where f , ..., f m , h , ..., h n are in L ( H, A dr ) and u , ..., u m , v , ..., v n are in Z ( e Φ ), we have ω ( θ, θ ′ ) := X i,j u i ◦ v j ◦ [ f i , h j ] . (5.4.5) Definition 5.4.2. The symplectic space related to the theory Φ and the renormalization map R ms is given by Γ := ( e Φ , Der Ham ( e Φ ) , ω ) . By choosing a Hamiltonian function F ∈ L ( H, A dr ), one can have a 0 − Hamiltonian system(Γ , F ) with respect to the theory and from proposition 5.3.28, a 0 − integral of motion for thissystem is an element f ∈ L ( H, A dr ) such that { f, F } = 0 . (5.4.6)Let θ F , θ f be the 0 − symplectic vector fields with respect to d F, d f . Therefore i θ F ω = d F, i θ f ω = d f, (5.4.7) { f, F } = i θ F i θ f ω = ω ( θ f , θ F ) = [ f, F ] = 0 . (5.4.8)47 orollary 5.4.3. Let f ∈ L ( H, A dr ) be a − integral of motion for the system (Γ , F ) . Then wehave R ms ◦ f ∗ R ms ◦ F = R ms ◦ F ∗ R ms ◦ f. Proof. By the formula (5.4.4), R ms ◦ [ f, F ] = [ R ms ◦ f, R ms ◦ F ] = ⇒ [ R ms ◦ f, R ms ◦ F ] = R ms (0) = 0 = ⇒ R ms ◦ f ∗ R ms ◦ F − R ms ◦ F ∗ R ms ◦ f = 0 . Corollary 5.4.4. The equation R ms ◦ f ∗ F − F ∗ R ms ◦ f + f ∗ R ms ◦ F − R ms ◦ F ∗ f − R ms ◦ ( f ∗ F ) + R ms ◦ ( F ∗ f ) = 0 determines a necessary and sufficient condition to characterize − integrals of motion for the − Hamiltonian system (Γ , F ) with respect to the minimal subtraction scheme in dimensionalregularization.Proof. It is proved by the equation (5.4.3). Because it shows that for a 0 − integral of motion f wehave [ f, F ] = 0 ⇐⇒ [ R ms ◦ f, F ] + [ f, R ms ◦ F ] − R ms ◦ [ f, F ] = 0 . If the Hamiltonian function of this system is the dimensionally regularized Feynman rulescharacter φ of the theory, then one can obtain interesting relations between the components ofthe Birkhoff factorization of φ and 0 − integrals of motion of this system. Corollary 5.4.5. For the dimensionally regularized Feynman rules character φ ∈ L ( H, A dr ) , let f be a − integral of motion for the − Hamiltonian system (Γ , φ ) . Then for each Feynman diagram Γ ∈ ker ǫ H , f satisfies in the equation X γ R ms ( f ( γ )) R ms ( φ ( Γ γ )) = X γ R ms ( φ ( γ )) R ms ( f ( Γ γ )) . Proof. For each Γ ∈ ker ǫ H , its coproduct is given by∆(Γ) = Γ ⊗ ⊗ Γ + X γ ⊂ Γ γ ⊗ Γ γ such that the sum is over all disjoint unions of 1PI superficially divergent proper subgraphs. Let f be a 0 − integral of motion of this system. Renormalization coproduct on Feynman diagramsshows us that ( R ms ◦ f ∗ R ms ◦ φ )(Γ) = R ms ( f (1)) R ms ( φ (Γ)) + R ms ( f (Γ)) R ms ( φ (1)) + X γ R ms ( f ( γ )) R ms ( φ ( Γ γ )) , ( R ms ◦ φ ∗ R ms ◦ f )(Γ) = R ms ( φ (Γ)) R ms ( f (1)) + R ms ( φ (1)) R ms ( f (Γ)) + X γ R ms ( φ ( γ )) R ms ( f ( Γ γ )) . Therefore with help of the corollary 5.4.3, the mentioned formula should be obtained such thatthe sum has finite terms where each term contains pole parts of Laurent series.48et f be a 0 − integral of motion for the system (Γ , φ ) such that φ is the Feynman rulescharacter of the theory. Theorem 4.2.6 shows that for each primitive 1PI (superficially divergent)proper subgraph γ of the Feynman diagram Γ, φ − ( γ ) = − R ms ( φ ( γ )) . (5.4.9)Therefore with the help of corollary 5.4.5 and (5.4.9), the relations between components of thefactorization of φ and this 0 − integral of motion can be available. We have X γ R ms ( f ( γ )) R ms ( φ ( Γ γ )) = X γ − φ − ( γ ) R ms ( f ( Γ γ ))= ⇒ X γ R ms ( f ( γ )) R ms ( φ ( Γ γ )) R ms ( f ( Γ γ )) − + φ − ( γ ) = 0= ⇒ X γ φ − ( γ ) = − X γ R ms ( f ( γ )) R ms ( φ ( Γ γ )) R ms ( f ( Γ γ )) − , (5.4.10) X γ φ + ( γ ) = X γ φ ( γ ) − X γ R ms ( f ( γ )) R ms ( φ ( Γ γ )) R ms ( f ( Γ γ )) − . (5.4.11) Corollary 5.4.6. (i) With applying (5.4.10) and (5.4.11) in the equations in theorem 4.2.6, newrepresentations from the factorization components of a Feynman rules character φ (based on the − integral of motion f ) can be determined.(ii) With putting them in (4.2.8) and with the help of lemma 4.2.7 and corollary 5.4.5, relationsbetween this − integral of motion and renormalization group and also β − function of the theorycan be investigated. Now we have a chance to search more geometrical meanings in the Feynman rules character φ and it can be performed based on motion integral condition for components of Birkhoff factoriza-tion of this character. It is necessary to emphasize that since the renormalization method is fixedand the factorization is unique, therefore consideration of this possibility helps us to familiar withsome more hidden geometrical structures in a quantum field theory which provide the requiredintegral conditions. Corollary 5.4.7. Suppose the negative part φ − of the Birkhoff decomposition of the dimensionallyregularized Feynman rules character φ ∈ L ( H, A dr ) of a given theory Φ be a − integral of motionfor the − Hamiltonian system (Γ , φ ) . Then for each Feynman diagram Γ , we have R ms ( X γ X γ ′ ⊂ Γ γ φ − ( γ ′ ) φ ( Γ γ γ ′ )) = 0 . Proof. If φ − be a 0 − integral of motion, then by given conditions in corollary 5.4.5 for each Feyn-man diagram Γ one should have X γ R ms ( φ − ( γ )) R ms ( φ ( Γ γ )) = X γ R ms ( φ ( γ )) R ms ( φ − ( Γ γ )) . By (5.4.9) and since R ms is an idempotent linear map, it can be seen that49 γ − R ms ( R ms ( φ ( γ ))) R ms ( φ ( Γ γ )) = X γ − R ms ( φ ( γ )) R ms ( φ ( Γ γ )) = X γ R ms ( φ ( γ )) R ms ( φ − ( Γ γ ))= ⇒ X γ − R ms ( φ ( Γ γ )) = X γ R ms ( φ − ( Γ γ )) = X γ φ − ( Γ γ ) . Set Γ γ := Γ γ . By theorem 4.2.6, X γ φ − (Γ γ ) = − R ms X γ ( φ (Γ γ ) + X γ ′ ⊂ Γ γ φ − ( γ ′ ) φ ( Γ γ γ ′ ))= ⇒ X γ R ms ( φ (Γ γ ) + X γ ′ ⊂ Γ γ φ − ( γ ′ ) φ ( Γ γ γ ′ )) = X γ R ms ( φ (Γ γ ))such that the sum is over all unions of 1PI proper superficially divergent subgraphs of Γ γ .With the help of theorems 4.2.5, 4.2.6 and the equation (5.4.4) and since R ms is an idempotentmap, one can obtain more explicitly conditions for the components of decomposition of φ whichplay the role of 0 − integrals of motion for the system (Γ , φ ). Proposition 5.4.8. For the − Hamiltonian system (Γ , φ ) such that φ is the dimensionally reg-ularized Feynman rules character (in L ( H, A dr ) ) of the theory Φ , the components of the Birkhofffactorization of φ are − integrals of motion for the system if and only if(i) φ − satisfies in the equation φ + − φ ∗ φ − + φ − ∗ R ms ◦ φ − R ms ◦ φ ∗ φ − + R ms ◦ ( φ ∗ φ − ) = 0 . (ii) φ + satisfies in the equation φ + ∗ R ms ◦ φ − R ms ◦ φ ∗ φ + − R ms ◦ ( φ + ∗ φ ) + R ms ◦ ( φ ∗ φ + ) = 0 . Proof. For the negative part φ − we have R ms ◦ φ − ∗ φ − φ ∗ R ms ◦ φ − + φ − ∗ R ms ◦ φ − R ms ◦ φ ∗ φ − − R ms ◦ ( φ − ∗ φ ) + R ms ◦ ( φ ∗ φ − ) = 0 ⇐⇒ φ − ∗ φ − φ ∗ φ − + φ − ∗ R ms ◦ φ − R ms ◦ φ ∗ φ − − R ms ◦ ( φ − ∗ φ ) + R ms ◦ ( φ ∗ φ − ) = 0 , φ = φ − − ∗ φ + ⇐⇒ φ + − φ ∗ φ − + φ − ∗ R ms ◦ φ − R ms ◦ φ ∗ φ − − R ms ◦ φ + + R ms ◦ ( φ ∗ φ − ) = 0 . For the positive part φ + we have R ms ◦ φ + ∗ φ − φ ∗ R ms ◦ φ + + φ + ∗ R ms ◦ φ − R ms ◦ φ ∗ φ + − R ms ◦ ( φ + ∗ φ ) + R ms ◦ ( φ ∗ φ + ) = 0 . Now since R ms ◦ φ + = 0 , the proof is completed.It is good place to consider the behavior of the renormalization group with respect to the0 − Hamiltonian system (Γ , φ ). With help of this group (associated to the Feynman rules character φ ), we are going to introduce an infinite dimensional 0 − integrable system with respect to the theoryΦ. 50 roposition 5.4.9. For each t , an element F t of the renormalization group is a − integral ofmotion for the − Hamiltonian system (Γ , φ ) if and only if it satisfies in F t ∗ R ms ◦ φ − R ms ◦ φ ∗ F t − R ms ◦ ( F t ∗ φ ) + R ms ◦ ( φ ∗ F t ) = 0 . Proof. Since for each t and Feynman diagram Γ, F t (Γ) is a polynomial in t , therefore R ms ( F t (Γ)) =0. With notice to (5.4.4), we have R ms ◦ F t ∗ φ − φ ∗ R ms ◦ F t + F t ∗ R ms ◦ φ − R ms ◦ φ ∗ F t − R ms ◦ ( F t ∗ φ ) + R ms ◦ ( φ ∗ F t ) = 0 . For arbitrary elements F t , F s of the renormalization group of each given renormalizable theoryΦ, one can observe that { F t , F s } = [ R ms ◦ F t , F s ] + [ F t , R ms ◦ F s ] − R ms ◦ ([ F t , F s ]) . (5.4.12)Since the renormalization group is a 1-parameter subgroup of G ( C ) (i.e. F t ∗ F s = F t + s ), thereforeit is easy to see that { F t , F s } = 0 . (5.4.13)It shows that the renormalization group can give us an integrable system and this fact turns outa strong relation between (the given Nijenhuis type) integrals of motion and the Connes-Kreimerrenormalization group. Corollary 5.4.10. For the renormalizable physical theory Φ , let φ ∈ G ( A dr ) be its dimensionallyregularized Feynman rules character and { F t } t be the renormalization group with respect to thischaracter. For each arbitrary element F t of the renormalization group, − Hamiltonain system (Γ , F t ) is an infinite dimensional − integrable system.Proof. The renormalization group contains infinite linearly independent 0 − integrals of motion F t for the system (Γ , F t ). Rosenberg’s strategy: The continuation of the standard process In [7] the authors (by working on the Lie group of diffeographisms and its related Lie algebra ofinfinitesimal characters) improve the study of infinite dimensional Lie algebras and factorizationproblem to the level of the Connes-Kreimer theory. In this part, we focus on their strategy andapply their results to consider factorization problem on the previously introduced noncommutativealgebras (deformed by Nijenhuis maps) such that consequently, a new family of integrals of motionwill be determined in a natural manner [92, 94].The Lie brackets [ ., . ] λ make available another procedure to study integrable systems at thislevel namely, identifying equations of motion and integral curves from a Lax pair equation. Set C drλ := ( L ( H, A dr ) , ◦ λ ) and supposing C λ = C drλ ⊕C dr ∗ λ be its related semisimple trivial Lie bialgebrasuch that C drλ := ( C drλ , [ ., . ] λ ). Corollary 5.5.1. C λ is the associated Lie algebra of the Lie group e C λ := C drλ ⋊ σ C dr ∗ λ such that σ : C drλ × C dr ∗ λ −→ C dr ∗ λ , ( f, X ) Ad ∗ ( f )( X ) . Proof. It is directly obtained from [7]. 51 efinition 5.5.2. The loop algebra of C λ is defined by the set L C λ := { F ( c ) = ∞ X j = −∞ c j F j , F j ∈ C λ } such that naturally, one can define the Lie bracket [ X c i F i , X c j G j ] := X k c k X i + j = k [ F i , G j ] λ on it. Decompose this set of formal power series into two parts L C λ, + = { ∞ X j =0 c j F j } , L C λ, − = { − X j = −∞ c j F j } (5.5.1)and let P ± are the natural projections on these components where P := P + − P − . Corollary 5.5.3. For a given Casimir function υ on L C λ , integral curve Λ( t ) of the Lax pairequation d Λ dt = [ M, Λ] where for F ( λ ) = F (0)( λ ) ∈ L C λ , M = P ( I ( dυ ( F ( c )))) ∈ L C λ is given by Λ( t ) = Ad ∗ L e C λ γ ± ( t ) . Λ(0) such that the smooth curves γ ± are the answers of the Birkhoff factorization exp ( − tX ) = γ − − ( t ) γ + ( t ) where X = I ( dυ ( F ( c ))) ∈ L C λ .Proof. It is easily calculated from [7, 86]. Remark 5.5.4. (i) It is important to know that one can project the above Lax pair equation toan equation on loop algebra of the original Lie algebra C drλ .(ii) Based on the induced Nijenhuis type symplectic structures, one can introduce a symplecticspace on the loop algebra L C λ . Next, depended motion integrals and therefore integrable Hamilto-nian systems on this loop algebra can be determined. Fixed point equations We observed that the Connes-Kreimer recalculation of the BPHZ renormalization depends stronglyon components of the Birkhoff factorization of dimensionally regularized Feynman rules charactersand on the other hand, the existence of this factorization is supported originally by Atkinson’stheorem [37]. Theorem 5.6.1. Let A be an associative algebra over the field K and R : A −→ A be a linear map.The pair ( A, R ) is a Rota-Baxter algebra if and only if the algebra A has a Birkhoff factorization.It means that there is a Cartesian product ( R ( A ) , e R ( A )) ⊂ A × A such that- It is a subalgebra of A × A ,- Each x ∈ A admits a unique decomposition x = R ( x ) ⊕ e R ( x ) . [2, 27] β − function and renormalization groupunderlying these equations will be considered [92, 94]. It should be important to note that becauseof the universality of the Connes-Kreimer Hopf algebra (with respect to the Hochschild cohomologytheory), the study of this family of fixed point equations at this level can be lifted to other Hopfalgebras of renormalizable theories. Lemma 5.6.1. Consider the group char A dr H x ( x = lrt, rt ) of characters on ladder trees (rootedtrees) with its related Lie algebra ∂ char A dr H x . This Lie algebra is generated by derivations Z t indexed by ladder tree (rooted tree) t and defined by the natural paring < Z t , s > = δ t,s . Based on the bijection between char A dr H x and ∂ char A dr H x , for each character g with thecorresponding derivation Z g , we have g = exp ∗ ( Z g ) = ∞ X n =0 Z ∗ ng n ! . (5.6.1) Lemma 5.6.2. (i) The Lie algebra ∂ char A dr H x together with the map R : f R ms ◦ f definea Lie Rota-Baxter algebra.(ii) Each derivation Z of this Lie algebra has a unique decomposition Z = R ( Z ) ⊕ e R ( Z ) . (iii) Idempotent property of R provides a decomposition A dr = A + dr ⊕ A − dr such that it can beextended to ∂ char A dr H x and it means that ∂ char A dr H x = ( ∂ char A dr H x ) + ⊕ ( ∂ char A dr H x ) − where ( ∂ char A dr H x ) + := e R ( ∂ char A dr H x ) , ( ∂ char A dr H x ) − := R ( ∂ char A dr H x ) [24, 25, 26]. For a fixed character g ∈ C xλ , let f be its integral of motion. Product ◦ λ , its related λ − Poissonbracket and motion integral condition show that { f, g } λ = i θ λg i θ λf ω λ = ω λ ( θ λf , θ λg ) = [ f, g ] λ = 0 . (5.6.2)And so it is apparently observed that[ R λ ( f ) , g ] + [ f, R λ ( g )] − R λ ([ f, g ]) = 0 ⇐⇒R λ ( f ) ∗ g − g ∗ R λ ( f ) + f ∗ R λ ( g ) − R λ ( g ) ∗ f − R λ ( f ∗ g ) + R λ ( g ∗ f ) = 0 . (5.6.3)53 roposition 5.6.3. For the given character g ∈ C xλ with the Birkhoff factorization ( g − , g + ) ,(i) g = g − − ∗ g + , R ms ◦ g − = g − , R ms ◦ g + = 0 . (ii) The negative component g − is an integral of motion for g iff g + − g ∗ g − + (1 + λ ) g − ∗ R ( g ) − (1 + λ ) R ( g ) ∗ g − + (1 + λ ) R ( g ∗ g − ) = 0 . (iii) The positive component g + is an integral of motion for g iff − λg + ∗ g + λg ∗ g + + (1 + λ ) g + ∗ R ( g ) − (1 + λ ) R ( g ) ∗ g + − (1 + λ ) R ( g + ∗ g ) + (1 + λ ) R ( g ∗ g + ) = 0 . Atkinson theorem determines very interesting recursive representation from components of de-composition of characters on rooted trees such that because of its practical structure in calculatingphysical information, this representation is called tree renormalization . Theorem 5.6.2. (i) Ladder tree renormalization. Each arbitrary character φ ∈ char A dr H lrt has a unique Birkhoff factorization ( φ − − , φ + ) such that φ = exp ∗ ( R ( Z φ ) + e R ( Z φ )) = exp ∗ ( R ( Z φ )) ∗ exp ∗ ( e R ( Z φ )) ,φ − = exp ∗ ( −R ( Z φ )) , φ + = exp ∗ ( e R ( Z φ )) . (ii) Rooted tree renormalization. Each arbitrary character ψ ∈ char A dr H rt has a uniqueBirkhoff factorization ( ψ − − , ψ ) such that ψ = exp ∗ ( Z ψ ) = exp ∗ ( R ( χ ( Z ψ ))) ∗ exp ∗ ( e R ( χ ( Z ψ ))) ,ψ − = exp ∗ ( −R ( χ ( Z ψ ))) , ψ + = exp ∗ ( e R ( χ ( Z ψ ))) where the infinitesimal character χ is characterized with the BCH series. [24, 25, 26, 27] Now with applying representations given in the theorem 5.6.2 and conditions given in theproposition 5.6.3, one can obtain new equations at the level of Lie algebra for while these compo-nents are motion integrals of a given character in the algebra C xλ .In addition, there is another procedure to deform the initial product (based on double Rota-Baxter structures) such that this new deformed product can induce another representation fromcomponents. Definition 5.6.4. Rota-Baxter map R deforms the convolution product ∗ to define a well knownassociative product on the set L ( H x , A dr ) given by f ∗ R g := f ∗ R ( g ) + R ( f ) ∗ g − f ∗ g. Lemma 5.6.5. (i) Information C x R := ( L ( H x , A dr ) , ∗ R , R ) is a Rota-Baxter algebra with thecorresponding R− bracket [ f, g ] R = [ f, R ( g )] + [ R ( f ) , g ] − [ f, g ] . (ii) For each infinitesimal character Z , it can be seen that exp ∗ ( R ( Z )) = R ( exp ∗ R ( Z )) , exp ∗ ( e R ( Z )) = − e R ( exp ∗ R ( − Z )) . Corollary 5.6.6. For the given character g ∈ C x R with the Birkhoff factorization ( g − , g + ) , thecomponents are integrals of motion for g iff(i) [ g − , R ( g )] = 0 ,(ii) [ g + , R ( g )] − [ g + , g ] = 0 , respectively.Proof. It is proved by theorem 5.6.2, definition 5.6.4 and lemma 5.6.5.54 orollary 5.6.7. Lie algebra version of the given equations in the corollary 5.6.6 are reformulatedby ( i ) R ( exp ∗ R ( − χ ( Z ψ ))) ∗ R ( exp ∗ ( Z ψ )) − R ( exp ∗ ( Z ψ )) ∗ R ( exp ∗ R ( − χ ( Z ψ ))) = 0 , ( ii ) − e R ( exp ∗ R ( − χ ( Z ψ ))) ∗ R ( exp ∗ ( Z ψ )) + R ( exp ∗ ( Z ψ )) ∗ e R ( exp ∗ R ( − χ ( Z ψ )))+ e R ( exp ∗ R ( − χ ( Z ψ ))) ∗ exp ∗ ( Z ψ ) − exp ∗ ( Z ψ ) ∗ e R ( exp ∗ R ( − χ ( Z ψ ))) = 0 . Now we want to relate a family of fixed point equations to the equations induced in the corollary5.6.7 and because of that we need a new operator. Definition 5.6.8. The morphism b [ ψ ] := exp ∗ R ( − χ ( Z ψ )) is called Bogoliubov character. Lemma 5.6.9. One can approximate Bogoliubov character with the formula R ( b [ ψ ]) = − R ms ◦ { exp ∗ ( Z ψ ) + α ψ } such that α ψ := X n ≥ n ! n − X j =1 n ! j !( n − j )! R ( − χ ( Z ψ )) ∗ ( n − j ) ∗ Z ∗ jψ . [26, 27] Corollary 5.6.10. One can rewrite equations in the corollary 5.6.7 based on the given estimationin the lemma 5.6.9. We have ( i ) ′ − R ( ψ + α ψ ) ∗ R ( exp ∗ ( Z ψ )) + R ( exp ∗ ( Z ψ )) ∗ R ( ψ + α ψ ) = 0 and ( ii ) ′ e R ( ψ + α ψ ) ∗ R ( exp ∗ ( Z ψ )) − R ( exp ∗ ( Z ψ )) ∗ e R ( ψ + α ψ ) − e R ( ψ + α ψ ) ∗ exp ∗ ( Z ψ )+ exp ∗ ( Z ψ ) ∗ e R ( ψ + α ψ ) = 0 . Furthermore, we saw that Birkhoff factorization of characters of the Connes-Kreimer Hopfalgebra are identified with the special infinitesimal character χ such that for each infinitesimalcharacter Z ∈ ∂char A dr H rt , it is given by χ ( Z ) = Z + ∞ X k =1 χ ( k ) Z . (5.6.4)The sum is a finite linear combination of infinitesimal characters and χ ( k ) Z s are determined bysolution of the fixed point equation E : χ ( Z ) = Z − ∞ X k =1 c k K ( k ) ( R ( χ ( Z )) , e R ( χ ( Z ))) (5.6.5)such that terms K ( k ) s are calculated with the BCH series [23, 24, 25, 26]. With putting theequation E in the Bogoliubov character and then applying the result 5.6.10, one can reformulatethe motion integral condition for components of a given character with respect to the fixed pointequation E . 55 roposition 5.6.11. For a given Feynman rules character ψ ∈ C rt R , if each of its Birkhofffactorization’s components is an integral of motion for ψ , then a class of fixed point equations willbe determined. Probably discussion about renormalization group and its infinitesimal generator (i.e. betafunction) can be interested at this level. We saw that these physical information are defined basedon the grading operator Y (that providing the scaling evolution of the coupling constant) suchthat this element exists from the extension of the Lie algebra ∂ char A dr H rt by an element Z where for each rooted tree t , we have[ Z , Z t ] = Y ( Z t ) = | t | Z t . (5.6.6) Lemma 5.6.12. For each character ψ ∈ char A dr H rt , its related β − function is given by β ( ψ ) = ψ − ∗ [ Z , ψ − − ] = ψ − ∗ Z ∗ ψ − − − Z . [26] It is obvious that with applying the exponential map, its related renormalization group isdetermined by F t = exp ∗ ( tβ ) . (5.6.7)On the other hand, we know that for each t ∈ R , F t is a character given by a polynomial of thevariable t and therefore R ms ◦ F t = 0 (5.6.8)([19, 25, 87]). Proposition 5.6.3 shows that Corollary 5.6.13. Each element of the renormalization group plays the role of an integral ofmotion for ψ in the algebras C xλ and C x R if and only if − λ [ F t , ψ ] + [ F t , R λ ( ψ )] − R λ ([ ψ, F t ]) = 0 , [ F t , R ( ψ )] − [ F t , ψ ] = 0 , respectively. The second condition in corollary 5.6.13 introduces a fixed point equation depended on theFeynman rules character ψ and its related β − function. Corollary 5.6.14. For a given Feynman rules character ψ ∈ C rt R , an element F t of the relatedrenormalization group plays the role of integral of motion for ψ iff the β − function satisfies in thefixed point equation [ exp ∗ ( tβ ) , R{ exp ∗ ( R ( E )) ∗ exp ∗ ( e R ( E )) } ] − [ exp ∗ ( tβ ) , exp ∗ ( R ( E )) ∗ exp ∗ ( e R ( E ))] = 0 . And finally, it should be remarked that because of the one parameter property of the renor-malization group (i.e. F t ∗ F s = F t + s ), one can easily show that for a fixed t in the cases C x and C x R , each F t is an integral of motion for F t . So it is reasonable to expect infinite integrablesystems related with algebra C x R . 56 hapter 6 Connes-Marcolli approach The Connes-Kreimer conceptional interpretation of renormalization theory could provide a newHopf algebraic reconstruction from physical information. In continuation of this approach, Connesand Marcolli initiated a new categorical framework based on geometric objects to describe renor-malizable physical theories and in fact, they showed that there is a fundamental mathematicalconstruction hidden inside of divergences. According to their programme, the BPHZ renormaliza-tion (i.e. minimal subtraction scheme in dimensional regularization) determines a principal bundlesuch that solutions of classes of differential equations related to some particular flat connectionson this bundle can encode counterterms. Then with collecting all of these connections into acategory, they introduced a new categorical formalism to consider physical theories such that itleaded to a universal treatment with respect to the Connes-Kreimer theory. [15, 16, 17, 18, 19]Universal singular frame is evidently one of the most important foundations of this machinerywhere it contains a deep physical meaning in the sense that with working at this level, all of thedivergences can be disappeared and so one can provide a finite theory. Moreover, one can findsignificance relations between this special frame and noncommutative geometry based on the localindex theorem. [15, 18]Nowadays a new theory of mixed Tate motives is developed on the basis of the Connes-Marcolliformalism which it reports about very desirable connections between theory of motives and quan-tum field theory. [15]In this chapter, we want to consider the basic elements of this categorical geometric foundationin the study of perturbative renormalization theory. Geometric nature of counterterms: Category of flat equi-singularconnections Reformulation of the Birkhoff decomposition in terms of classes of differential equations determinessome geometric objects with respect to physical information of renormalizable theories. In fact,negative parts of decomposition of loops (or characters) can be applied to correct the behaviorof solutions of these differential systems by flat connections (together with a special singular-ity, namely equi-singularity) near the singularities, without making more singularities elsewhere.The importance of this possibility can be investigated in making a geometrically encoding fromcounterterms (divergences) [11, 12, 16, 17, 18, 19]. In this part an overview from this story isdone. Definition 6.1.1. For a given connected graded commutative Hopf algebra H of finite type withthe related complex Lie group G ( C ) and Lie algebra g ( C ) , let α : I = [ a, b ] ⊂ R −→ g ( C ) be a mooth curve. Its associated time ordered exponential is defined by T e R ba α ( t ) dt := 1 + X n ≥ Z a ≤ s ≤ ... ≤ s n ≤ b α ( s ) ...α ( s n ) ds ...ds n such that the product is taken in the graded dual space H ∗ and ∈ H ∗ is the unit correspondingto the counit ǫ . Remark 6.1.2. (i) This integral only depends on 1-form α ( t ) dt and because of finite type propertyof Hopf algebra, for each element in H , it is finite with values in the affine group scheme.(ii) Time ordered exponential is the value g ( b ) of the unique solution g ( t ) ∈ G ( C ) of thedifferential equation dg ( t ) = g ( t ) α ( t ) dt with the initial condition g ( a ) = 1 .(iii) It is multiplicative over the sum of paths.(iv) Let ω be a flat g ( C ) − valued connection on Ω ⊂ R and α : [0 , ⊂ R −→ Ω be a curve.Then the time ordered exponential T e R α ⋆ ω only depends on the homotopy class [ α ] of paths suchthat α (0) = a, α (1) = b .(v) Let ω ∈ g ( C ( { z } )) has a trivial monodromy M ( ω ) = 1 . Then there exists a solution g ∈ G ( C ( { z } )) for the equation D ( g ) = ω such that the logarithmic derivative D : G ( K ) −→ Ω ( g ) is given by D ( g ) := g − dg . [15, 16, 17, 18, 19] Time ordered exponential necessarily and sufficiently provides an useful representation fromloops on the set Loop ( G ( C ) , µ ) and their Birkhoff components such that it contains a reformu-lation of loops in terms of a class of differential systems associated to a family of equi-singularflat connections. As the consequence, this machinery delivers us a geometric meaningful fromdivergences. Theorem 6.1.1. Let γ µ ( z ) be a loop in the class Loop ( G ( C ) , µ ) . Then(i) There exists a unique β ∈ g ( C ) and a loop γ reg ( z ) regular at z = 0 such that γ µ ( z ) = T e − z R − zlogµ ∞ θ − t ( β ) dt θ zlogµ ( γ reg ( z )) . (ii) There is a representation from components of the Birkhoff factorization of γ µ ( z ) based ontime ordered exponential. We have γ µ + ( z ) = T e − z R − zlogµ θ − t ( β ) dt θ zlogµ ( γ reg ( z )) ,γ − ( z ) = T e − z R ∞ θ − t ( β ) dt . (iii) For each element β ∈ g ( C ) and regular loop γ reg ( z ) , one unique element of Loop ( G ( C ) , µ ) is identified. [15, 16, 17, 18, 19] Definition 6.1.3. Define an equivalence relation on connections (i.e. g ( C ) − valued one forms). ω , ω are equivalent if there exists h ∈ G ( C { z } ) such that ω = D h + h − ω h. Remark 6.1.4. There is a correspondence between two equivalence g ( C ) − valued connections ω , ω with trivial monodromies and negative parts of the Birkhoff decomposition of solutions ofthe equations D f = ω , D f = ω where f , f ∈ G ( C ( { z } )) . It means that ω ∼ ω ⇐⇒ f − = f − . imensional regularization is an usual regularization technique in perturbative renormaliza-tion. It is based on an analytic continuation of Feynman integrals to the complex dimension d ∈ C in a neighborhood of the integral (critical) dimension D at which ultra-violet divergencesoccur. Under this regularization prescription, the procedure of renormalization with a specialrenormalization scheme namely, minimal subtraction (i.e. the subtraction of singular part of theLaurent series in z = d − D at each order in the loop expansion) can be performed. In the BPHZrenormalization, all of the (sub-)divergences, in a recursive procedure, disappear such that it canbe understood in terms of the extraction of finite values (i.e. Birkhoff decomposition of loops withvalues in the space of diffeographisms). Theorem 6.1.2. There is a principal bundle connected with a given theory under the Dim. Reg.+ Min. Sub. scheme such that one special class of its related connections provides a geometricreinterpretation from physical information. This bundle is called renormalization bundle. [16, 17,18, 19]Proof. A sketch of proof: Let ∆ be an infinitesimal disk corresponds with the complexified di-mension D − z ∈ ∆ of dimensional regularization and G m ( C ) = C ∗ be the possible choices for thenormalization of an integral in dimension D − z . For the principal C ∗ − bundle G m −→ B −→ ∆ ,let P = B × G be a trivial principal G − bundle. Set V := p − ( { } ) ⊂ B, B := B − V, P = B × G. This bundle, together with connections on it, can store some geometrical meanings related tophysical information. Remark 6.1.5. The choice of the unit of mass µ is the same as the choice of a section σ : ∆ −→ B and indeed, the concept of equi-singularity turns to this fact. Definition 6.1.6. Fix a base point y ∈ V and identify B with ∆ × G m ( C ) . A flat connection ω on P is called equi-singular, if- It is G m ( C ) − invariant,- For any solution f of the equation D f = ω , the restrictions of f to the sections σ : ∆ −→ B with σ (0) = y have the same singularity (namely, the pullbacks of a solution have the samenegative parts of the Birkhoff decomposition, independent of the choice of the section and thereforethe mass parameter). It is remarkable that one can expand equivalence relation given by definition 6.1.3 to thiskind of connections. With working on classes of equi-singular flat connections on renormalizationprincipal bundle, one can redisplay infinitesimal characters. Theorem 6.1.3. Equivalence classes of flat equi-singular G − connections on P are representedby elements of g ( C ) and also, each element of the Lie algebra of affine group scheme of theHopf algebra associated to the renormalizable theory Φ identifies a specific class of equi-singularconnections. The above process is independent of the choice of a local regular section σ : ∆ −→ B with σ (0) = y . [16, 17, 18, 19] Theorems 6.1.1, 6.1.3 provide a bijective correspondence between negative parts of the Birkhoffdecomposition of loops with values in G ( C ) and classes of equi-singular flat connections on therenormalization bundle (which stores the regularization parameter). Since counterterms of thetheory are identified with these negative parts (i.e. Connes-Kreimer formalization), thereforethese classes of connections are presenting the counterterms.In a categorical configuration, one can introduce a category such that equi-singular flat connec-tions are its objects. The construction of this category and its properties are completely analyzedby Connes and Marcolli and just we review their main result.59 heorem 6.1.4. For a given renormalizable QFT Φ underlying Dim. Reg. + Min. Sub. withthe associated Lie group G ( C ) and vector bundle B −→ ∆ , flat equi-singular connections on thisbundle introduce an abelian tensor category with a specific fiber functor. Additionally, this categoryis a neutral Tannakian category and therefore it is equivalent to the category of finite dimensionalrepresentations of affine group scheme G ∗ := G ⋊ G m . [15, 16, 17, 18, 19] The construction of a universal Tannakian category The Riemann-Hilbert correspondence conceptually consists of describing a certain category ofequivalence classes of differential systems though a representation theoretic datum. For a givenrenormalizable QFT Φ with the related Hopf algebra H and the space of diffeographisms G ( C ),it was shown that how one can identify a category of classes of flat equi-singular G − connections.Categorification of the Connes-Kreimer theory allows us to have a reasonable idea to formulatethis story in a universal setting (underlying the Riemann-Hilbert problem) by constructing theuniversal category E of equivalence classes of all flat equi-singular vector bundles.This category has the ability of covering the corresponding categories of all renormalizabletheories and it means that when we are working on the theory Φ, it is possible to considerthe subcategory E Φ of those flat equi-singular vector bundles which is equivalent to the finitedimensional linear representations of G ∗ . In this part we try to consider some general features ofthis very special category and its universality.Start with a filtered vector bundle ( E, W ) over B with an increasing filtration W − n − ( E ) ⊂ W − n ( E ) , ( W − n ( E ) = M m ≥ n E m ) (6.2.1)and set Gr Wn = W − n ( E ) W − n − ( E ) . (6.2.2) Definition 6.2.1. A W − connection on E is a connection ▽ on the vector bundle E = E | B such that- ▽ is compatible with the filtration,- ▽ is the trivial connection on Gr Wn . Remark 6.2.2. One can extend the mentioned equivalence relation in the definition 6.1.3 andthe concept of equi-singularity to the level of W − connections. It means that(i) Two W − connections ▽ , ▽ on E are W − equivalent if there exists an automorphism T of E that preserves the filtration, where is identity on Gr Wn and T ◦ ▽ = ▽ ◦ T .(ii) A flat W − connection ▽ on E is equi-singular if it is G m ( C ) − invariant and the pullbackalong different sections σ of B (such that σ (0) = y ) of a solution of the equation ▽ η = 0 havethe same type of singularity. Definition 6.2.3. The couple ( E, ▽ ) is called a flat equi-singular vector bundle. These pairs introduce a category E such that its objects Obj ( E ) are data Θ = [ V, ▽ ] where V is a Z − graded finite dimensional vector space and ▽ is an equi-singular W − connection on thefiltered bundle E = B × V (with attention to the classes of connections).Each morphism T ∈ Hom (Θ , Θ ′ ) is a linear map T : E −→ E ′ compatible with the gradingsuch that it should have the following relation with the connections ▽ , ▽ ′ . Set ▽ := (cid:0) ▽ ′ ▽ (cid:1) , ▽ := (cid:0) ▽ ′ T ◦ ▽ − ▽ ′ ◦ T ▽ (cid:1) . (6.2.3)60or the defined connections in (6.2.3) on the vector bundle ( E ′ L E ) ⋆ , ▽ should be a conjugateof ▽ by the unipotent matrix (cid:0) T (cid:1) . Theorem 6.2.1. With applying Riemann-Hilbert correspondence, one can formulate Connes-Marcolli category (consisting of divergences of a renormalizable theory) in a universal configura-tion by constructing the universal category E of equivalence classes of all flat equi-singular vectorbundles with the the fiber functor given by ϕ : E −→ V C , Θ V . [16, 17, 18, 19, 76] We know that there is a representation of a neutral Tannakian category by the category R G ∗ offinite dimensional representations of the affine group scheme of automorphisms of the fiber functorof the main category. This fact provides a new reformulation from this specific universal category. Theorem 6.2.2. E is a neutral Tannakian category. It is equivalent to the category R U ∗ of finitelinear representations of one special affine group scheme (related to the universal Hopf algebra ofrenormalization H U ) namely, universal affine group scheme U ∗ such that H U is a connected gradedcommutative non-cocommutative Hopf algebra of finite type. It is the graded dual of the universalenveloping algebra of the free graded Lie algebra L U := F (1 , , ... ) • generated by elements e − n ofdegree n > (i.e. one generator in each degree). [17, 19] Remark 6.2.4. (i) Since the category of filtered vector spaces is not an abelian category, it isnecessary to use the direct sum of bundles and the above condition on connections.(ii) The construction of this category over the field Q is also possible. With attention to theorem 6.1.3 and also given correspondence in theorem 6.2.2, one can obtaina new prescription from equi-singular connections such that it will be applied to produce a newuniversal level for counterterms, renormalization groups and β − functions of theories. Theorem 6.2.3. Let H be a connected graded commutative Hopf algebra of finite type with theaffine group scheme G and e P := B × G ∗ . Each equivalence class ω of flat equi-singular con-nections on e P identifies a graded representation ρ ω : U ∗ −→ G ∗ (which is identity on G m ) andalso, for each graded representation ρ : U −→ G , there is one specific class of flat equi-singularconnections. [16, 18] Based on the correspondence R U ∗ ≃ E , for each object Θ = [ V, ▽ ], there exists a uniquerepresentation ξ Θ of U ∗ in V such that D ξ Θ ( γ U ) ≃ ▽ . And also each arbitrary representation ξ of U ∗ in V determines a unique connection ▽ (up to equivalence) such that [ V, ▽ ] is an objectin E . It can be seen that one specific element will be determined from the process namely, theloop universal singular frame γ U with values in U ( C ) where at this level one hopes to eliminatedivergences for generating a finite theory.Now because of the equivalence relation between loops (with values in the Lie group G ( C ))and elements of the Lie algebra (corresponding to G ( C )), with the help of a suitable element ofthe Lie algebra L U , one can characterize the universal singular frame. Lemma 6.2.5. The Lie algebra element corresponding to the universal singular frame is e = P n ≥ e − n (i.e. the sum of the generators of the Lie algebra). Since the universal Hopf algebra of renormalization is finite type, whenever we pair e with anelement of the Hopf algebra, it will be only a finite sum. Hence e makes sense. Theorem 6.2.4. (i) e is an element in the completion of L U .(ii) e : H U −→ K [ t ] is a linear map. Its affine group scheme level namely, rg : G a ( C ) −→ U ( C ) is a morphism that plays an essential role to calculate the renormalization group.(iii) The universal singular frame can be reformulated with γ U ( z, v ) = T e − z R v u Y ( e ) duu .(iv) For each loop γ ( z ) in Loop ( G ( C ) , µ ) , with help of the associated representation ρ : U −→ G ,the universal singular frame γ U maps to the negative part γ − ( z ) of the Birkhoff decomposition of γ ( z ) and also, the renormalization group { F t } t in G ( C ) can be recalculated by ρ ◦ rg . [16, 18] G m − principal bundle B .Positive and negative components of the Birkhoff factorization of the loop γ U in the pro-unipotent affine group scheme U contain a universal meaning. For a given renormalizable theoryΦ these components, via the identified representations in the theorem 6.2.3, map to renormalizedvalues and counterterms, respectively. This fact provides a valuable concept namely, universalcounterterms .At last, it is favorable to emphasize the universality of the category E among other categoriesdetermined from physical theories. It means that this category gives us the power of analyzingflat equi-singular connections for each affine group scheme and this is the main reason of itsuniversality. Theorem 6.2.5. Let H be a connected graded commutative Hopf algebra of finite type with theaffine group scheme G . Let ω be a flat equi-singular connection on P and ψ : G −→ GL ( V ) afinite dimensional linear graded representation of G . We can correspond an element Θ ∈ Obj ( E ) to the data ( ω, ψ ) . The equivalence class of ω identifies the same element Θ . [16, 17, 19] Remark 6.2.6. For each flat equi-singular vector bundle ( E, ▽ ) (such that E = B × V ), theconnection ▽ identifies a flat equi-singular G V − valued connection ω . Indeed, ▽ is given by addinga Lie G V − valued one form to the trivial connection. This correspondence preserves the mentionedequivalence relation in the definition 6.1.3. In summary, it is observed that lifting the Connes-Kreiemr perturbative renormalization to theuniversal configuration can yield the particular Hopf algebra H U . Because of the combinatorialnature of the universal Connes-Kreimer Hopf algebra, it should be reasonable to search a hiddencombinatorial construction in the backbone of this Hopf algebra. This problematic notion is themain topic of the next chapter and its importance will be investigated when we want to generalizethe Connes-Marcolli treatment in the study of Dyson-Schwinger equations.62 hapter 7 Universal Hopf algebra ofrenormalization We discussed that how one can systematically interpret the hidden combinatorics of perturbativerenormalization based on the Hopf algebra structure on Feynman diagrams of a pQFT. Further-more, it was exhibited that the Connes-Kreimer Hopf algebra of rooted trees (equipped withsome decorations which represent primitive 1PI graphs) plays the role of an available practicalmodel such that with changing labels, it will upgrade for each arbitrary theory. On the otherhand, we saw that the universal affine group scheme U governs the structure of divergences of allrenormalizable theories and the universality of H U turns to its independency from all theories.In this chapter, we are going to discover the natural combinatorial backbone of H U by givinga new explicit rooted tree type reformulation from this particular Hopf algebra and its related Liegroup. Then with using this new interpretation, first we obtain rigorous relations between universalHopf algebra of renormalization and some well-known combinatorial Hopf algebras. Secondly, weexpand this new Hall tree formalism to the level of Lie groups and finally, we will describe theuniversal singular frame based on Hall polynomials such that as the result, new Hall tree scatteringformulaes for physical information will be determined. [93] Shuffle nature of H U The Hopf algebra H U is introduced in the theorem 6.2.2 and in particular, one important note isthat as an algebra it is isomorphic to the linear space of noncommutative polynomials in variables f n , n ∈ N > with the shuffle product. It is a skeleton key for us to find a relation between thisHopf algebra and rooted trees and because of that at first we need some more information aboutshuffle structures. Definition 7.1.1. Let V be a vector space over the field K of characteristic zero and T ( V ) = L n ≥ V ⊗ n be its related tensor algebra. Set S ( m, n ) = { σ ∈ S m + n : σ − (1) < ... < σ − ( m ) , σ − ( m + 1) < ... < σ − ( m + n ) } . It is called the set of ( m, n ) − shuffles. For each x = x ⊗ ... ⊗ x m ∈ V ⊗ m , y = y ⊗ ... ⊗ y n ∈ V ⊗ n and σ ∈ S ( m, n ) , define σ ( x ⊗ y ) = u σ (1) ⊗ u σ (2) ⊗ ... ⊗ u σ ( m + n ) ∈ V ⊗ ( m + n ) such that u k = x k for ≤ k ≤ m and u k = y k − m for m + 1 ≤ k ≤ m + n . The shuffle product of , y is given by x ⋆ y := X σ ∈ S ( m,n ) σ ( x ⊗ y ) . Lemma 7.1.2. ( T ( V ) , ⋆ ) is a unital commutative associative algebra. [23, 72] There are some extensions of this product such as quasi-shuffles, mixable shuffles. Let A be a locally finite set (i.e. a disjoint union of finite sets A n , n ≥ A are letters andmonomials are called words such that the empty word is denoted by 1. Set A − := A ∪ { } . Definition 7.1.3. Function < ., . > : A − × A − −→ A − is called a Hoffman pairing, if it satisfiesfollowing conditions:- For all a ∈ A − , < a, > = 0 ,- For all a, b ∈ A − , < a, b > = < b, a > ,- For all a, b, c ∈ A − , << a, b >, c > = < a, < b, c >> ,- For all a, b ∈ A − , < a, b > = 0 or | < a, b > | = | a | + | b | .A locally finite set A together with a Hoffman pairing < ., . > on A − is called a Hoffman set. Definition 7.1.4. Let K < A > be the graded noncommutative polynomial algebra over K . Thequasi-shuffle product ⋆ − on K < A > is defined recursively such that for any word w , ⋆ − w = w ⋆ − w and also for words w , w and letters a, b , ( aw ) ⋆ − ( bw ) = a ( w ⋆ − bw ) + b ( aw ⋆ − w )+ < a, b > ( w ⋆ − w ) . Lemma 7.1.5. K < A > together with the product ⋆ − is a graded commutative algebra such thatwhen < ., . > = 0 , it will be the shuffle algebra ( T ( V ) , ⋆ ) where V is a vector space generated bythe set A . [23, 39, 72] Shuffle type products can determine interesting family of Hopf algebras such that it can bepossible to reformulate H U on the basis of this class. The next theorem gives a complete charac-terization from this class of Hopf algebras. Theorem 7.1.1. (i) The (quasi-) shuffle product introduces a graded connected commutative non-cocommutative Hopf algebra structure (of finite type) on ( K < A >, ⋆ − ) and ( T ( V ) , ⋆ ) .(ii) There is an isomorphism (as a graded Hopf algebras) between ( T ( V ) , ⋆ ) and ( K < A >, ⋆ − ) .(iii) There is a graded connected Hopf algebra structure (of finite type) (comes from (quasi-)shuffle product) on the graded dual of K < A > .(iv) We can extend the isomorphism in the second part to the graded dual level. [23, 39, 72]Proof. The compatible Hopf algebra structure on the shuffle algebra of noncommutative polyno-mials is given by the coproduct ∆( w ) = X uv = w u ⊗ v and the counit ǫ (1) = 1 , ǫ ( w ) = 0 , w = 1 . For a given Hoffman pairing < ., . > and any finite sequence S of elements of the set A , withinduction define < S > ∈ A − such that for any a ∈ A , < a > = a, < a, S > = < a, < S >> . Let C ( n ) be the set of compositions of n and C ( n, k ) be the set of compositions of n with length k . For each word w = a ...a n and composition I = ( i , ..., i l ), set I < w > := < a , ..., a i >< a i +1 , ..., a i + i > ... < a i + ... + i l − +1 , ..., a n > . 64t means that compositions act on words. Now for any word w = a ...a n , its antipode is given by S (1) = 1 ,S ( w ) = − n − X k =0 S ( a ...a k ) ⋆ − a k +1 ...a n = ( − n X I ∈ C ( n ) I < a n ...a > . The isomorphism between these Hopf algebra structures (compatible with the shuffle products)is given by morphisms τ ( w ) = X ( i ,...,i l ) ∈ C ( | w | ) i ! ...i l ! ( i , ..., i l ) < w >,ψ ( w ) = X ( i ,...,i l ) ∈ C ( | w | ) ( − | w |− l i ...i l ( i , ..., i l ) < w > . The graded dual K < A > ⋆ has a basis consisting of elements v ⋆ (where v is a word on A )with the following pairing such that if u = v , then ( u, v ⋆ ) = 1 and if u = v , then ( u, v ⋆ ) = 0. ItsHopf algebra structure is given by the concatenation product conc ( u ⋆ ⊗ v ⋆ ) = ( uv ) ⋆ and the coproduct δ ( w ⋆ ) = X u,v ( u ⋆ − v, w ⋆ ) u ⋆ ⊗ v ⋆ . The map τ ⋆ ( u ⋆ ) = X n ≥ X = u n ! ( a ...a n ) ⋆ , is an isomorphism in the dual level and its inverse is given by ψ ⋆ ( u ⋆ ) = X n ≥ ( − n − n X = u ( a ...a n ) ⋆ . Because of the relation between Hopf algebras and Lie theory, one can consider shuffle Hopfalgebras at the Lie algebra version. Definition 7.1.6. Let L be a Lie algebra over K . There exists an associative algebra L over K together with a Lie algebra homomorphism φ : L −→ L such that for each couple ( A , φ : L −→A ) of an algebra and a Lie algebra homomorphism, there is a unique algebra homomorphism φ A : L −→ A such that φ A ◦ φ = φ . L is called universal enveloping algebra of L and it isunique up to isomorphism. Lemma 7.1.7. (i) Universal enveloping algebra L of the free Lie algebra L ( A ) is a free associativealgebra on A .(ii) φ is an injective morphism such that φ ( L ( A )) is the Lie subalgebra of L generated by φ ◦ i ( A ) . [82] Definition 7.1.8. The set of Lie polynomials in K < A > ⋆ is the smallest sub-vector space of K < A > ⋆ containing the set of generators A ⋆ := { a ⋆ : a ∈ A } and closed under the Lie bracket. orollary 7.1.9. (i) The set of Lie polynomials in K < A > ⋆ forms a Lie algebra. It is the freeLie algebra on A ⋆ such that K < A > ⋆ is its universal enveloping algebra.(ii) In the shuffle product case, the Lie polynomials are exactly the primitives for δ and thereforeat the level of (quasi-)shuffle product, the primitives are elements of the form ψ ⋆ p such that p isa Lie polynomial [39]. Now it is the place to come back to the definition of the universal Hopf algebra of renormaliza-tion. It can be seen that the set A = { f n : n ∈ N > } is a locally finite set and as an algebra, H U is isomorphic to ( T ( V ) , ⋆ ) such that V is a vector space over C spanned by the set A . Thereforeits Hopf algebra structure is determined with the theorem 7.1.1. At the Lie algebra level, we haveto go to the dual structure. Corollary 7.1.9 shows us that the set of all Lie polynomials in H ⋆ U isthe free Lie algebra generated by { f ⋆n } n ∈ N > such that H ⋆ U is its universal enveloping algebra andon the other hand, we know that H ⋆ U is identified by the universal enveloping of the free gradedLie algebra L U generated by { e − n } n ∈ N > . Corollary 7.1.10. This procedure implies to have a reasonable correspondence between generatorsof the Lie algebra L U and elements of the set A ⋆ . Rooted tree version In this part, with attention to the shuffle nature of H U , we introduce a combinatorial versionfrom this specific Hopf algebra in the Connes-Marcolli universal renormalization theory [93]. Weconsider completely a new Hall rooted tree reformulation from H U to obtain interesting relationsbetween this particular Hopf algebra and some well-known combinatorial Hopf algebras introducedin [4, 14, 39, 40, 41, 42, 43, 44, 70, 71, 81, 86, 98]. Moreover, we extend this formalization of H U to the level of its associated Lie group to display its elements based on formal series of Hall trees.So it provides a new Hall tree type representation from universal singular frame γ U where we willconsider the applications of Hall basis and PBW basis to reformulate combinatorially physicalinformation. Definition 7.2.1. Defining a partial order (cid:22) on the set of all rooted trees T . We say t (cid:22) s , if t can be obtained from s by removing some non-root vertices and edges and it implies that | t | ≤ | s | . Definition 7.2.2. Let T ( A ) ( F ( A ) ) be the set of all rooted trees (forests) labeled by the set A .(i) For a ∈ A , t , ..., t m ∈ T ( A ) such that u = t ...t m ∈ F ( A ) , B + a ( u ) is a labeled rooted treeof degree | t | + ... + | t m | + 1 obtained by grafting the roots of t , ..., t m to a new root labeled by a .In addition, B + a ( I ) is a rooted tree with just one labeled vertex.(ii) For t ∈ T ( A ) and u ∈ F ( A ) , define a new element t ◦ u such that it is a labeled rooted treeof degree | t | + | u | given by grafting the roots of labeled rooted trees in u to the root of t . Lemma 7.2.3. (i) The operation ◦ is not associative.(ii) ∀ t ∈ T ( A ) , ∀ u, v ∈ F ( A ) : ( t ◦ u ) ◦ v = t ◦ ( uv ) = ( t ◦ v ) ◦ u .(iii) t ◦ ... ◦ t m ◦ u = t ◦ ( t ◦ ... ◦ ( t m ◦ u )) , t ◦ k = t ◦ ... ◦ t , k times.(iv) For each u ∈ F ( A ) , let per ( u ) be the number of different permutations of the vertices of alabeled partially ordered set that representing u . Then per ( I ) = 1 , per ( B + a ( u )) = per ( u ) . And if u = Q mj =1 ( t j ) i j , then per ( u ) = m Y j =1 i j ! per ( t j ) i j . (v) The bilinear extension of ◦ to the linear combinations of labeled rooted trees (or linearcombinations of labeled forests) is also possible. efinition 7.2.4. A set H ( T ( A )) of labeled rooted trees is called Hall set, if it has followingconditions:- There is a total order relation > on H ( T ( A )) .- If a ∈ A , then B + a ( I ) ∈ H ( T ( A )) .- For a ∈ A , u ∈ F ( A ) − { I } such that u = t ◦ r ...t ◦ r m m , t , ..., t m ∈ H ( T ( A )) , r , ..., r m ≥ , t > ... > t m , B + a ( u ) ∈ H ( T ( A )) ⇐⇒ t m > B + a ( t ◦ r ...t ◦ r m − m − ) ∈ H ( T ( A )) . - If t = B + a ( t ◦ r ...t ◦ r m m ) ∈ H ( T ( A )) such that t , ..., t m ∈ H ( T ( A )) , r , ..., r m ≥ , a ∈ A , thenfor each j = 1 , ..., m , t j > t . Lemma 7.2.5. For each t ∈ H ( T ( A )) , r ≥ and a ∈ A , it is easy to see that B + a ( t ◦ r ) ∈ H ( T ( A )) ⇐⇒ t > B + a ( I ) . Definition 7.2.6. For a Hall set H ( T ( A )) , the set of its forests is given by H ( F ( A )) := { I } ∪ { t r ...t r m m : r , ..., r m ≥ , t , ..., t m ∈ H ( T ( A )) , t i = t j ( i = j ) } . Lemma 7.2.7. Elements of H ( F ( A )) and rooted trees are in the relation with the map ξ : H ( F ( A )) − { I } −→ T ( A ) t ◦ r ...t ◦ r m m −→ t ◦ r ◦ ( t ◦ r ...t ◦ r m m ) .ξ is injective and its image is the set { B + a ( u ) ∈ T ( A ) : u ∈ H ( F ( A )) , a ∈ A } . [73] Remark 7.2.8. (i) Hall trees and Hall forests have no symmetry.(ii) There is a one to one correspondence between a Hall set of A − labeled rooted trees and aHall set of words on A . [82] Definition 7.2.9. For t ∈ H ( T ( A )) , there is a standard decomposition ( t , t ) ∈ H ( T ( A )) × H ( T ( A )) such that- If | t | = 1 , then the decomposition is t = t, t = I ,- And if t = B + a ( t ◦ r ...t ◦ r m m ) such that r , ..., r m ≥ , t , ..., t m ∈ H ( T ( A )) : t > ... > t m ,a ∈ A , then the decomposition is given by t = B + a ( t ◦ r ...t ◦ r m − m − t ◦ r m − m ) , t = t m . - For a Hall forest u ∈ H ( F ( A )) − H ( T ( A )) such that u = t ◦ r ...t ◦ r m m , t > ... > t m , thedecomposition is given by ( u , u ) ∈ H ( F ( A )) × H ( T ( A )) where u = t ◦ r ...t ◦ r m − m − t ◦ r m − m , u = t m . Definition 7.2.10. For a given map that associates to each word w on A a scalar α w ∈ K , definea map α : F ( A ) −→ K such that- I α .- For each u ∈ F ( A ) − { I } , there is a labeled partially ordered set ( u ( A ) , ≥ ) that representsthe forest u such that vertices x , ..., x n , ... of this poset are labeled by l ( x i ) = a i ∈ A ( ≤ i ).Let > u ( A ) be a total order relation on the set of vertices u ( A ) such that it is an extension of thepartial order relation ≥ on u ( A ) . For each ordered sequence x i > u ( A ) ... > u ( A ) x i n in u ( A ) , itscorresponding word a i ...a i n is denoted by w ( > u ( A ) ) . Set α ( u ) := X > u ( A ) α w ( > u ( A ) ) , where the sum is over all total order relations > u ( A ) (i.e. extensions of the main partial orderrelation ≥ ) on the set of vertices of u ( A ) . emma 7.2.11. Define a map π given by π : K [ T(A) ] −→ ( K < A >, ⋆ − ) , π ( u ) := X > u ( A ) w ( > u ( A ) ) . One can show that for each u, v ∈ F(A) and a ∈ A ,(i) π ( B + a ( u )) = π ( u ) a, (ii) π ( uv ) = π ( u ) ⋆ − π ( v ) , (iii) α ( u ) = b α ( π ( u )) , where b α : ( K < A >, ⋆ − ) −→ K , b α ( w ) = α w is a K − linear map. Definition 7.2.12. For given maps α, β : F ( A ) −→ K , one can define a new map αβ : F ( A ) −→ K given by u X ( v ( A ) , w ( A )) ∈ R ( u ( A )) α ( v ) β ( w ) such that u ( A ) is a labeled poset representing u and v ( A ) , w ( A ) are labeled partially ordered subsetsof u ( A ) such that the set of all pairs ( v ( A ) , w ( A )) with the following conditions is denoted by R ( u ( A )) .- The set of vertices in u ( A ) is the disjoint union of the set of vertices v ( A ) and w ( A ) ,- For each x, y ∈ u ( A ) such that x ≥ y , if x ∈ w ( A ) then y ∈ w ( A ) . Remark 7.2.13. It can be seen that for each word w = a ...a m on A , ( αβ ) w = α w β + α β w + m − X j =1 α a ...a j β a j +1 ...a m . Definition 7.2.14. For the map α given by the definition 7.2.10, one can introduce an equivalencerelation on K [ T(A) ] such that for each u, v ∈ K [ T(A) ] , they are congruent ( u ≡ v ), if for everymap b α : A ∗ −→ K , w α w , (such that A ∗ is the set of all words on A ), then we have α ( u ) = α ( v ) . Remark 7.2.15. (i) u ≡ v ⇐⇒ u − v ∈ ker π .(ii) u, v ∈ K [ T(A) ] , a ∈ A , u ≡ v = ⇒ B + a ( u ) ≡ B + a ( v ) .(iii) u, v ∈ F(A) : u ≡ v = ⇒ uu ≡ vv .(iv) t ∈ T(A) , n ≥ t n ≡ n ! t ◦ n .(v) t ∈ T(A) , i, j ≥ t ◦ i t ◦ j ≡ ( i + j )! i ! j ! t ◦ ( i + j ) . Lemma 7.2.16. For m ≥ and t , ..., t m ∈ T(A) , with induction one can show that t ...t m ≡ m X i =1 t i ◦ Y j = i t j . There is an algorithm (in finite number of recursion steps) for rewriting each u ∈ F(A) as u ≡ v such that v ∈ K H ( F ( A )) (i.e. v is a K − linear combination of Hall forests). [73] Definition 7.2.17. There is a canonical map f on Hall rooted trees defined by- f ( a ) = a , if a ∈ A ,- f ( t ) = f ( t ) f ( t ) , if t be of degree ≥ with the standard decomposition t = ( t , t ) .The function f is called foliage and for each Hall tree t , its degree | f ( t ) | is the number of leavesof t . The foliage of a Hall tree is called Hall word. Theorem 7.2.1. For each word w on A , there is a unique factorization w = f ( t ) ...f ( t n ) suchthat t i ∈ H ( T ( A )) and t > ... > t n . [82] One can show that Hall sets of A -labeled rooted trees can be reconstructed recursively froman arbitrary Hall set of words on A . It means that68 orollary 7.2.18. A Hall set of words on A is the image under the foliage of a Hall set H ( T ( A )) of labeled rooted trees. There is an important class of words (i.e. Lyndon words) such that one can deduce Hallsets from them and moreover, this kind of words can store interesting information about shufflestructures. Let us start with a well-known order. Definition 7.2.19. Let A be a totally ordered set. The alphabetical ordering determines a totalorder on the set of words on A such that for any nonempty word v , put u < uv and also for letters a < b and words w , w , w , put w aw < w bw . Definition 7.2.20. For a given total order set A , a non-trivial word w is called Lyndon, if forany non-trivial factorization w = uv , we have w < v . The first advantage of these words can be seen in their influencing role in making a Hall set. Theorem 7.2.2. The set of Lyndon words, ordered alphabetically, is a Hall set. [39, 82] Theorem 7.2.3. Let A be a locally finite set equipped with a total order relation. The (quasi-)shuffle algebra ( K < A >, ⋆ − ) is the free polynomial algebra on the Lyndon words. [39] It is shown that the universal Hopf algebra of renormalization as an algebra is defined by theshuffle product on the linear space of noncommutative polynomials with variables f n ( n ∈ N ).This determines an important order. Definition 7.2.21. With referring to given correspondence in the result 7.1.10, one can define anatural total order relation (depending on the degrees of the generators e − n , ( n ∈ N ) of the freeLie algebra L U ) on the set A = { f n : n ∈ N > } . It is given by f m > f n ⇐⇒ n > m. Theorem 7.2.3 shows that H U (as an algebra) is the free polynomial algebra of Lyndon words onthe set A and therefore one can consider Hall set of these Lyndon words (ordered alphabetically)such that its corresponding Hall set of labeled rooted trees is denoted by H ( T ( A )) U . It can beseen obviously that the set of Lyndon words is an influencing factor in determining this bridgebetween rooted trees and H U .It is near to have our interesting rooted tree reformulation. Let us consider free commutativealgebra K [ T(A) ] such that the set { t r ...t r m m : t , ..., t m ∈ T(A) } is a K − basis (as a graded vectorspace) where each expression t r ...t r m m is a forest. Definition 7.2.22. For the forest u with the associated partial order set ( u ( A ) , ≥ ) , define acoproduct given by ∆( u ) = X ( v ( A ) , w ( A )) ∈ R ( u ( A )) v ⊗ w such that labeled forests v, w are represented by labeled partially ordered subsets v ( A ) , w ( A ) of u ( A ) . Theorem 7.2.4. Coproduct 7.2.22 determines a connected graded commutative Hopf algebrastructure on K [ T(A) ] such that the product in the dual space K [ T(A) ] ⋆ = { α : T(A) −→ linear K } corresponds to the fixed coproduct namely, dual of the coalgebra structure and it means that foreach α, β ∈ K [ T(A) ] ⋆ and each forest u , αβ ( u ) = ( α ⊗ β )∆( u ) . [72, 73] emark 7.2.23. One can show that H GL (labeled by the set A ) and K [ T(A) ] are graded dual toeach other. By using theorem 7.2.4 and operation B + a , one can show that this Hopf algebra has a universalproperty. Theorem 7.2.5. Let H be a commutative Hopf algebra over K and { L a : H −→ H } a ∈ A be afamily of K − linear maps such that ∪ a ∈ A ImL a ⊂ kerǫ H and ∆ H L a ( c ) = L a ( c ) ⊗ I H + ( id H ⊗ L a )∆ H ( c ) . Then there exists a unique Hopf algebra homomorphism ψ H : K [ T(A) ] −→ H such that for each u ∈ K [ T(A) ] and a ∈ A , we have ψ H ( B + a ( u )) = L a ( ψ H ( u )) . [72, 73] Corollary 7.2.24. Theorem 7.2.5 is a poset version of the combinatorial Connes-Kreimer Hopfalgebra and it means that H CK (labeled by the set A ) is isomorphic to K [ T(A) ] .Proof. It is clearly proved based on the theorem 3.1.2 (universal property of H CK ). Lemma 7.2.25. There is a bijection between the set of non-empty words and the set of labeledrooted trees without side-branchings.Proof. One can show that the K − linear map π is a Hopf algebra homomorphism and for each a , ..., a m ∈ A , we have π ( B + a m ...B + a ( a )) = a ...a m . Corollary 7.2.26. The map π is an epimorphism and for each b α, b β ∈ K < A > ⋆ , u ∈ K [ T(A) ] ,we have < b α b β, π ( u ) > = < αβ, u > such that α, β ∈ K [ T(A) ] ⋆ . Everything is ready to introduce a rooted tree version of shuffle type Hopf algebras. Theorem 7.2.6. (i) The (quasi-)shuffle Hopf algebra ( K < A >, ⋆ − ) is isomorphic to the quotientHopf algebra K [ T(A) ] I π such that I π := Kerπ is a Hopf ideal in K [ T(A) ] with the generators < { Q mi =1 t i − P mi =1 t i ◦ Q j = i t j : m > , t , ..., t m ∈ T(A) } > = < { t ◦ z + z ◦ t − tz : t, z ∈ T(A) } ∪ { s ◦ t ◦ z + s ◦ z ◦ t − s ◦ ( tz ) : t, z, s ∈ T(A) } > = < { t ◦ z + z ◦ t − tz : t, z ∈ T(A) } ∪ { s ◦ ( tz ) + z ◦ ( ts ) + t ◦ ( sz ) − tzs : t, z, s ∈ T(A) } > . (ii) As an K − algebra, K [ T(A) ] I π is freely generated by the set { t + I π : t ∈ H ( T ( A )) } . [72, 73] With attention to the given operadic picture from Connes-Kreimer Hopf algebra in the pastchapters, next result can be indicated immediately.70 orollary 7.2.27. For each locally finite set A together with a total order relation, there existHopf ideals J , J such that ( K < A >, ⋆ − ) ∼ = K [ T(A) ] I π ∼ = H CK ( A ) J ∼ = H NAP ( A ) J . So universal Hopf algebra of renormalization is isomorphic to a quotient of the (labeled) incidenceHopf algebra with respect to the basic set operad NAP.Proof. By theorems 3.3.3, 7.2.5, it is enough to set J := I π and J := ρ − J . In addition, theorem7.2.6 is in fact a representation of H U by rooted trees. Because it is enough to replace the set A with the variables f n such that the identified Lyndon words (with the shuffle structure of H U )gives us the Hall set H ( T ( A )) U .In continue of this part, we use this new redefinition from universal Hopf algebra of renormal-ization to provide some interesting relations between this Hopf algebra and other combinatorialHopf algebras. Proposition 7.2.28. Rooted tree reformulation of H U determines the following commutative di-agrams of Hopf algebra homomorphisms. N SY M β −−−−→ H Fβ y β y SY M β −−−−→ H U SY M β ⋆ ←−−−− U ( L U ) β ⋆ y β ⋆ y QSY M β ⋆ ←−−−− H P (7.2.1) Proof. It is easy to see that SY M ⊂ QSY M . As a vector space, QSY M is generated by the mono-mial quasi-symmetric functions M I such that I = ( i , ..., i k ) and M I := P n Holtkamp in [44] proved that the Foissy Hopf algebra H F is isomorphic to theHopf algebra C [ Y ∞ ] of planar binary trees ([70]) and moreover, Foissy in [31] found an isomor-phism between H F and photon Hopf algebra H γ ([4]) (related to renormalization). With the help ofthese facts and proposition 7.2.28, one can find new homomorphisms between these Hopf algebrasand H U . Now we want to lift the Zhao’s homomorphism given in the definition 3.2.8 and its dual tothe level of the universal Hopf algebra of renormalization. It is shown that π is a surjectivehomomorphism from H CK ( A ) to H U . On the other hand, Z ⋆ provides a unique surjective mapfrom H CK to QSY M with the property 3.2.9. For a word w with length n in H U , there exists alabeled ladder tree l wn of degree n in H CK ( A ) such that π ( l wn ) = w . Definition 7.2.30. Define a new map Z u : H U −→ QSY M such that for each element w ∈ H U ,it is given by Z u ( w ) := Z ⋆ ( l wn ) . It can be seen that Z u is a homomorphism of Hopf algebras and it is unique with respectto the relation 3.2.9. With the help of theorem 3.2.4 and proposition 7.2.28, one can definehomomorphisms θ : N SY M −→ H U , θ := β ◦ β = β ◦ β , (7.2.11) θ : H GL −→ H U , θ := β ◦ α ⋆ = β ◦ α ⋆ . (7.2.12) Lemma 7.2.31. The surjective morphism π induces a new homomorphism Ξ from H CK to H U such that for each unlabeled forest u in H CK , it is defined by Ξ( u ) := X v ∈ [ u ] π ( v ) . Proposition 7.2.32. Rooted tree reformulation of H U determines the following commutative di-agram. SY M α w w ppppppppppp β (cid:15) (cid:15) H CK Ξ & & NNNNNNNNNNN H GLα ⋆ g g NNNNNNNNNNN θ x x pppppppppppp H U Z u x x ppppppppppp QSY M N SY M α w w ppppppppppp θ f f NNNNNNNNNNNN SY M Z u ◦ β g g NNNNNNNNNNN β O O (7.2.13)It is also possible to consider the dual version of the above diagram.73 emma 7.2.33. Dual of the maps Z u and Ξ can be give by Z ⋆u : N SY M −→ U ( L U ) , Z ⋆u ( z n ) := e − n , Ξ ⋆ : U ( L U ) −→ H GL , Ξ ⋆ ( e − n ) := l n . Proposition 7.2.34. The dual version of the diagram (7.2.13) is given by SY M α ' ' NNNNNNNNNNN H GL α ⋆ ooooooooooo H CK U ( L U ) β ⋆ O O β ⋆ (cid:15) (cid:15) Ξ ⋆ f f NNNNNNNNNNN θ ⋆ & & NNNNNNNNNNN θ ⋆ ppppppppppp N SY M Z ⋆u ppppppppppp ( Z u ◦ β ) ⋆ ' ' OOOOOOOOOOO QSY MSY M α ⋆ ppppppppppp (7.2.14)Based on the graded dual relation, there is another procedure to obtain other connectionsbetween H U and combinatorial Hopf algebras. Lemma 7.2.35. Let H and H be graded connected locally finite Hopf algebras which admit innerproducts ( ., . ) and ( ., . ) , respectively. If they are dual to each other, then there is a linear map λ : H −→ H such that(i) λ preserves degree,(ii) For each h , h ∈ H : ( h , h ) = ( λ ( h ) , λ ( h )) , (iii) For each h , h , h ∈ H :( h h , h ) = ( λ ( h ) ⊗ λ ( h ) , ∆ ( λ ( h ))) , ( h ⊗ h , ∆ ( h )) = ( λ ( h ) λ ( h ) , λ ( h )) . [41] Remark 7.2.36. Lemma 7.2.35 determines an isomorphism τ : H −→ H ⋆ such that for each h ∈ H and h ∈ H , it is defined by < τ ( h ) , h > := ( h , λ ( h )) . Since H CK ( A ) and H GL ( A ) are graded dual to each other, therefore by 7.2.35 one can finda linear map λ : H CK ( A ) −→ H GL ( A ) with the mentioned properties such that it defines anisomorphism τ from H GL ( A ) to H CK ( A ) ⋆ given by < τ ( t ) , s > := ( t, λ ( s )) (7.2.15)where for rooted trees t , t , if t = t then ( t , t ) = | sym ( t ) | and otherwise it will be 0. For eachword w with length n in H U , there is a labeled ladder tree l wn in H CK ( A ) such that π ( l wn ) = w . Proposition 7.2.37. There is a homomorphism of Hopf algebras F : H U −→ H GL ( A ) given by F ( w ) := τ − (( l wn ) ⋆ ) . 74e know that H U and U ( L U ) are graded dual to each other. And therefore lemma 7.2.35 intro-duces a linear map θ : H U −→ U ( L U ) with the mentioned properties. This morphism determinesan isomorphism τ from U ( L U ) to H ⋆ U such that for each element x ∈ U ( L U ) and word w ∈ H U ,we have < τ ( x ) , w > := ( x, θ ( w )) (7.2.16)where ( ., . ) is the natural pairing on U ( L U ). Proposition 7.2.38. For a labeled forest u with the corresponding element π ( u ) in H U , based onthe natural pairing (given in theorem 7.1.1), the dual of F is clarified by F ⋆ : H CK ( A ) −→ U ( L U ) ,F ⋆ ( u ) := τ − (( π ( u )) ⋆ ) . It should be remarked that one can extend this rooted tree reformulation from H U to its relatedaffine group scheme U where it is done by the theory of construction of a group from an operad[83]. Let P be an augmented set operad and K A P = L n K ( P n ) S n be the direct sum of its relatedcoinvariant spaces with the completion [K A P = Q n K ( P n ) S n . Lemma 7.2.39. There is an associative monoid structure on [K A P . [14, 86] Let G P be the set of all elements of [K A P whose first component is the unit . It is a subgroupof the set of invertible elements. One can generalize this notion to a categorical level. Theorem 7.2.7. There is a functor from the category of augmented operads to the category ofgroups. [14, 86] Naturally, one can expect a Hopf algebra based on subgroup G P . Theorem 7.2.8. For a given operad P , we have a commutative Hopf algebra structure on K [ G P ] given by the set of coinvariants of the operad. [14, 86]Proof. It is a free commutative algebra of functions on G P generated by the set ( g α ) α ∈ A P . Each f ∈ G P can be represented by a formal sum f = P α ∈ A P g α ( f ) α such that g = 1.Relation with incidence Hopf algebra is the first essential property of this Hopf algebra. It canbe seen that there is a surjective morphism η : g α F [ α ] | Aut ( α ) | (7.2.17)from the Hopf algebra K [ G P ] to the incidence Hopf algebra H P . Further, based on this morphismit is observed that at the level of groups, the Lie group G P ( K ) of H P is a subgroup of the group G P . [14, 98] Corollary 7.2.40. (i) There is a surjective morphism from the Hopf algebra K [ G NAP ] to theConnes-Kreimer Hopf algebra H CK of rooted trees.(ii) The complex Lie group G ( C ) of H CK is a subgroup of G NAP .Proof. With referring to theorem 3.3.3 and (7.2.17), the map ρ ◦ η : K [ G NAP ] −→ H CK is asurjective morphism and in terms of groups we will get the second claim.For the operad N AP , G NAP is a group of formal power series indexed by the set of unlabeledrooted trees and there is also an explicit picture from the elements of G NAP ( C ) such that one canlift it to the level of universal Hopf algebra of renormalization.75 orollary 7.2.41. (i) Group U ( C ) is a subgroup of G NAP and therefore each of its elements canbe represented by a formal power series indexed with Hall rooted trees.(ii) Formal series K = P t g t ( K ) t in G NAP belongs to U ( C ) if and only if- t be a Hall tree in H ( T ( A )) U , (It does not belong to the Hopf ideal I π .)- And if t = B + f n ( u ) for some u = t ...t k such that t , ..., t k ∈ H ( T ( A )) U and f n ∈ A , one has g t ( K ) = k Y i =1 g B + fn ( t i ) ( K ) . Proof. One can extend the morphism (7.2.17) to the level of the decorated Hopf algebras andtherefore by corollaries 7.2.27 and 7.2.40, there is a surjective map from C [ G NAP ]( A ) to H U . Soit shows that in terms of groups, U ( C ) is a subgroup of G NAP . For the second case, according tothe lemma 6.12 in [14], each element of G NAP is in the subgroup G NAP ( C ) if and only if for eachtree t = B + ( t , ..., t k ), we have the following condition | sym ( t ) | g t ( K ) = k Y i =1 | sym ( B + ( t i )) | g B + ( t i ) ( K ) . By theorems 3.3.17, 3.3.3, corollaries 7.2.27, 7.2.40 and (7.2.17), and since Hall trees have nosymmetries, the proof is completed. Remark 7.2.42. The universal affine group scheme U ∗ := U ⋊ G m has a manifestation as amotivic Galois group for a category of mixed Tate motives. Suppose Q ( ζ N ) be the cyclotomic fieldof level N and O be its ring of integers. For N = 3 or , the motivic Galois group of the category MT mix ( S N ) of mixed Tate motives where S N = Spec ( O [ N ]) is non-canonically of the form U ∗ .Moreover for the scheme S N = Spec ( Z [ i ][ ]) of N − cyclotomic integers, one can find a non-canonical isomorphism between U ∗ and the motivic Galois group G M T ( Z [ i ][ ]) [18, 19, 36, 35].So with applying corollaries 7.2.40 and 7.2.41 one can translate this Hall tree formalism to thesemotivic Galois groups (at the level of groups) to obtain a new Hall tree type representation fromtheir elements. And finally, let us have a remark about Lie algebra version. It will help us to characterizePBW basis and Hall basis with respect to the Hall set H ( T ( A )) U . Definition 7.2.43. Let H ( A ) be a Hall set and A ⋆ := { a ⋆ : a ∈ A } . For each Hall word w , itsassociated Hall polynomial p w in the free Lie algebra L ( A ⋆ ) is introduced by- If f k j ∈ A , then p f kj = f ⋆k j ,- If w be a Hall word of length ≥ such that its corresponding Hall tree t w has the standarddecomposition ( t w , t w ) , then p w = [ p w , p w ] . Lemma 7.2.44. (i) With induction one can show that each p w is a homogeneous Lie polynomialof degree equal to the length of w and also, it has the same partial degree with respect to each letteras w .(ii) For a given Hall set H , Hall polynomials form a basis for the free Lie algebra (viewed as avector space) and their decreasing products p f k ...p f kn such that f k i ∈ H, f k > f k > ... > f k n ,form a basis for the free associative algebra (viewed as a vector space) [82]. We found a Hall set H ( T ( A )) U corresponding to H U such that A = { f n } n ∈ N > and for each f n its associated Hall polynomial is given by p f n = e − n . (7.2.18)The next result completes our Hall tree reconstruction.76 orollary 7.2.45. (i) As a vector space, Hall polynomials associated to the Hall set H ( T ( A )) U form a basis for the Lie algebra L U .(ii) As a vector space, decreasing products of Hall polynomials with respect to the Hall set H ( T ( A )) U form a basis for the free algebra H U . Therefore in summary, based on the basic set operad NAP on rooted trees, one can pro-vide a complete reconstruction from the specific Hopf algebra H U in the algebraic perturbativerenormalization, its related infinite dimensional complex Lie group and also Lie algebra (withnotice to the Milnor-Moore theorem). Now it can be useful to apply this induced version in theConnes-Marcolli’s universal treatment such that consequently, new reformulations from physicalinformation such as counterterms can be expected. This project is actually performed by universalsingular frame. Universal singular frame The appearance of the Connes-Marcolli categorification of renormalization theory depends directlyupon the Riemann-Hilbert problem such that as the result one can search the application of thetheory of motives in describing divergences. Neutral Tannakian formalism of the category of flatequi-singular vector bundles could determine the special loop γ U with values in the universal affinegroup scheme U ( C ). Since now we have enough knowledge about the combinatorics of the universalHopf algebra of renormalization, so it does make sense to search for a combinatorial nature insideof the universal singular frame. The explanation of this important frame (free of any divergences)based on Hall basis and PBW basis is the main purpose of this section such that universality of γ U allows us to modify this new Hall type representation to counterterms of any renormalizabletheory. [93]Starting with the application of words in geometry. Let M be a smooth manifold, C ( R + ) thering of real valued piecewise continuous functions on R + and { X a } a ∈ A a family of smooth vectorfields on M . Suppose A be the algebra over C ( R + ) of linear operators on C ∞ ( M ) generated bythe vector fields X a ( a ∈ A ). For a family { g a } a ∈ A of elements in C ( R + ), set X ( x ) = X a ∈ A g a ( x ) X a . (7.3.1) Lemma 7.3.1. The formal series (7.3.1) can be expanded as a series of linear operators in A ofthe form P w g w X w such that- w = a ...a m is a word on A ,- X = Id (identity operator), X w = X a ...X a m ,- g w = R a m ... R a C ( R + ) where each R a i : C ( R + ) −→ C ( R + ) , (1 ≤ i ≤ m ) is a linear endomor-phism defined by { Z a i g } ( x ) := Z x g ( s ) g a i ( s ) ds. This technique can be generalized to any arbitrary algebra. Theorem 7.3.1. For a given associative algebra A over the field K of characteristic zero generatedby the elements { E a } a ∈ A , all elements in A are characterized by formal series P w µ w E w suchthat µ w ∈ K . [73, 72, 82] Remark 7.3.2. If A is a free algebra, then this formal series representation is unique. Definition 7.3.3. Suppose a be the unital Lie algebra generated by the set { E a } a ∈ A . For a givenHall set H ( T ( A )) of labeled rooted trees with the corresponding Hall forest H ( F ( A )) , one canassign elements E ( u ) such that E ( I ) = e (The unit of a ),- For each Hall tree t with the standard decomposition ( t , t ) ∈ H ( T ( A )) × H ( T ( A )) , E ( t ) = [ E ( t ) , E ( t )] = E ( t ) E ( t ) − E ( t ) E ( t ) , - For each u ∈ H ( F ( A )) − H ( T ( A )) with the standard decomposition ( u , u ) ∈ H ( F ( A )) × H ( T ( A )) , E ( u ) = E ( u ) E ( u ) . Lemma 7.3.4. (i) The Lie algebra a is spanned by { E ( t ) : t ∈ H ( T ( A )) } (as an ordered basis).It is called Hall basis.(ii) A is spanned by { E ( u ) : u ∈ H ( F ( A )) } . It is called PBW basis. [72, 73, 82] It is the place to give a new understanding from the universal singular frame. Proposition 7.3.5. For the locally finite total order set { f n } n ∈ N , the universal singular framecan be rewritten by γ U ( − z, v ) = X n ≥ ,k j > α Uf k f k ...f kn p f k ...p f kn v P k j z − n such that p f kj s are Hall polynomials.Proof. From theorem 6.2.4, we know that γ U ( z, v ) = T e − z R v u Y ( e ) duu . After the application of the time ordered exponential, one can get γ U ( − z, v ) = X n ≥ ,k j > e − k ...e − k n k ( k + k ) ... ( k + ... + k n ) v P k j z − n such that in this expansion the coefficient of e − k ...e − k n is calculated by the iterated integral Z ≤ s ≤ ... ≤ s n ≤ s k − ...s k n − n ds ...ds n . On the other hand, theorem 7.2.3 determines that Hopf algebra H U is a free polynomial algebraon Lyndon words on the set { f n } n ∈ N > . Consider formal series E := f k + xf k + x f k + ... where µ k j ( x ) = x k j − . By the lemma 7.3.1, for the variables 0 ≤ s ≤ ... ≤ s n ≤ 1, we have { Z k j } ( s j ) = Z s j x k j − dx. For each word f k f k ...f k n , we can define the following well-defined iterated integral α U f k f k ...f kn := Z k n ... Z k . It is easy to see that the above integral is agree with the iterated integral associated to thecoefficient of the term e − k ...e − k n . This completes the proof.78n addition, one can determine uniquely a real valued map on the set F ( A ). Definition 7.3.6. Let A = { f n } n ∈ N be the locally finite total order set corresponding to theuniversal Hopf algebra of renormalization. For the given map in the above proof that associatesto each word w = f k f k ...f k n a real value α U w and based on the definition 7.2.10, we introduce anew map α U on F ( A ) such that- I α U ( I ) = 1 , - For each non-empty labeled forest u in F ( A ) , α U ( u ) = X > u ( A ) α U w ( > u ( A ) ) . Lemma 7.3.7. For given labeled rooted trees t , ..., t m ∈ T ( A ) and f k j ∈ A , α U ( t ...t m ) = α U ( t ) ...α U ( t m ) , α U ( B + f kj ( t ...t m )) = Z k j α U ( t ) ...α U ( t m ) . The map α U (determined by the universal singular frame), with the above properties, can beuniquely identified. Lemma 7.3.8. For the given map in definition 7.3.6 together with the mentioned properties inlemma 7.3.7, there exists a real valued map β U on F ( A ) given by α U = exp β U such that for each u ∈ F ( A ) − T ( A ) , β U ( u ) = 0 .Proof. (A sketch of the proof.) For a given map α : F ( A ) −→ K ,If α ( I ) = 0, then the exponential map is defined by exp α ( I ) = 1 , and for each u ∈ F ( A ) − { I } , exp α ( u ) = | u | X k =1 k ! α k ( u ) . And if α ( I ) = 1, then the logarithm map is defined by log α ( I ) = 0 , and for each u ∈ F ( A ) − { I } , log α ( u ) = | u | X k =1 ( − k +1 k ( α − ǫ ) k ( u )such that ǫ ( I ) = 1 and for u ∈ F ( A ) − { I } , ǫ ( u ) = 0. With the help of proposition 7 in [73], theproof is completed. Proposition 7.3.9. X k j > ,n ≥ α U f k f k ...f kn f k f k ...f k n = exp ( X t ∈ H ( T ( A )) U β U ( t ) E ( t )) such that- { E ( t ) : t ∈ H ( T ( A )) U } (i.e. the set of all Hall polynomials) is the Hall basis for L U ,- { E ( u ) : u ∈ H ( F ( A )) U } (i.e. the set of decreasing products of Hall polynomials) is the PBWbasis for H U . roof. One can show that for a given map α : F ( A ) −→ K , if α ( I ) = 0, then exp ( X w α w E w ) = X u ∈ H ( F ( A )) exp α ( u ) E ( u ) , and if α ( I ) = 1, then log ( X w α w E w ) = X u ∈ H ( F ( A )) log α ( u ) E ( u ) . Now by lemma 7.3.7 and with help of the continuous BCH formula (44) in [73], for each word w = f k f k ...f k n on the set A , one can have exp ( X t ∈ H ( T ( A )) U β U ( t ) E ( t )) = X w α U w w. Hall basis and PBW basis depended on H U can be determined by corollary 7.2.45 and lemma7.3.4.With attention to lemma 7.3.4, definition 7.3.6 and proposition 7.3.5, a Hall representationfrom γ U is obtained. Definition 7.3.10. The formal series P t ∈ H ( T ( A )) U β U ( t ) E ( t ) becomes a Hall polynomial repre-sentation for the universal singular frame. Remark 7.3.11. There is also another way to show that the universal singular frame has a rootedtree representation. Corollary 7.2.40 and (7.2.17) give us a surjective morphism from C [ G NAP ]( A ) to the Hopf algebra H U . And also corollary 7.2.41 shows that the infinite dimensional Lie group U ( C ) is a subgroup of G NAP . Since γ U is a loop with values in U ( C ) , therefore for each fixedvalues z, v , γ U ( z, v ) should be a formal power series of Hall rooted trees with the given conditionsin corollary 7.2.40. Now let us come to conclusion. In [25, 26] one can find rooted tree type reformulations ofcomponents of the Birkhoff decomposition of a dimensionally regularized Feynman rules characterbased on Baker-Campbell-Hausdorff (BCH) series and Bogoliubov character such that as theresult, scattering formulaes for the renormalization group and the β − function can be investigated.Now here we can obtain a new Hall rooted tree scattering type formula for counterterms based onthe given combinatorial version of H U . Corollary 7.3.12. One can map the Hall polynomial representation of the universal singularframe to counterterms of an arbitrary pQFT.Proof. One can show that for each loop γ µ ( z ) in Loop ( G ( C ) , µ ) (where µ is the mass parameter),with help of the graded representation ξ γ µ : U −→ G identified by theorem 1.106 in [19], theuniversal singular frame γ U maps to the negative part γ − ( z ) of the Birkhoff decomposition of γ µ ( z ). And we know that this minus part does independent of µ and it determines counterterms.Now it is enough to map Hall trees formulaes given by propositions 7.3.5 and 7.3.9 underlying ξ γ µ .Since Hall set H ( T ( A )) U and its related Hall polynomials determine Hall basis and PBWbasis, one can expect to reproduce the BCH type representations of physical information (givenin [25, 26]) with respect to this Hall tree approach. This possibility reports another reason for theimportance of H U in the study of renormalizable theories. Proposition 7.3.13. On the basis of formal sums of Hall trees (Hall forests) and Hall polynomialsassociated to the universal Hopf algebra of renormalization, one can introduce new reformulationsfrom divergences (counterterms) of each arbitrary renormalizable theory and furthermore, it canbe applied to redefine the renormalization group and its infinitesimal generator. hapter 8 Combinatorial Dyson-Schwingerequations and Connes-Marcolliuniversal treatment The study of Dyson-Schwinger equations (DSEs) could help people as an influenced method ofdescribing unknown non-perturbative circumstances and in fact, it has central role in the develop-ment of modern physics. These equations enable us to find an effective conceptional explanationfrom non-perturbative theory which is the main complicate part of quantum field theory. It seemsthat a non-perturbative theory can be discovered with solving its corresponding DSE. The signif-icance of this approach for the general identification of quantum field theory is on the increase.[69, 85]In modern physics people concentrate on an analytic approach to this type of practical equa-tions such that it contains a class of equations related to ill-defined iterated Feynman integrals.More interestingly, Kreimer introduces a new combinatorial reinterpretation from this class ofequations such that in his strategy, he systematically applies the combinatorics of renormalizationtogether with Hochschild cohomology theory to obtain perturbative expansions of Hochschild onecocycles depended upon Hopf algebra of Feynman diagrams. This process justifies combinatorialDyson-Schwinger equations (DSEs) and so it will be considerable to derive non-perturbative the-ory from the capsulate renormalization Hopf algebra. It should be mentioned that in principle,this formalism is strongly connected with the analytic formulation. Because Kreimer could regainanalytic DSEs from this combinatorial version by using one particular measure which relates Feyn-man rules characters (in the Hopf algebraic language) with their corresponding standard forms.[5, 6, 51, 53, 54, 56, 57, 58]In short, on the one hand, DSEs report non-perturbative phenomena and on the other hand,they can be formalized on the basis of infinite formal perturbative expansions of one cocycles. So itmeans that the formalization of quantum field theory underlying Hopf algebra of renormalizationleads us widely to clarify a more favorable explanation from non-perturbative quantum field theory.In this chapter at first we consider Kreimer’s programme in the study of DSEs based onHochschild cohomology theory on various Hopf algebras of rooted tress. Then after redefiningone cocycles at the Lie algebra level, we consider DSEs at the level of the universal Hopf algebraof renormalization (with attention to its described combinatorics). In the next step, we wantto show the importance of H U in the theory of DSEs and this work will be done based on thefactorization principal of Feynman diagrams into primitive components. As the result, we willextend reasonably the universality of H U (i.e. independency of all renormalizable physical theories)to non-perturbative theory. And finally, we show that the study of DSEs at the level of the81niversal Hopf algebra of renormalization hopefully leads to a new explanation from this kind ofequations in a combinatorial-categorical configuration. This procedure provides one importantfact that the universality of the category of equi-singular flat vector bundles can be developed toDSEs and therefore non-perturbative theory. [91, 96, 97]Totally, these observations mean that beside analytic and combinatorial techniques, one canexpect a new categorical geometric interpretation from this family of equations. Hopf algebraic reformulation of quantum equations of motion basedon Hochschild cohomology theory One crucial property of DSEs is their ability in describing the self-similarity nature of amplitudesin renormalizable QFTs such that it can help us to introduce a recursive combinatorial versionfrom these equations. Starting with the physical theory Φ with the associated Hopf algebra H F G = H (Φ). With applying the renormalization coproduct, one can indicate a coboundaryoperator which introduces a Hochschild cohomology theory with respect to the Hopf algebra H F G . Definition 8.1.1. Define a chain complex ( C, b ) with respect to the coproduct structure of H suchthat the set of n − cochains and the coboundary operator are determined with C n := { T : H −→ H ⊗ n : Linear } , b T := ( id ⊗ T )∆ + n X i =1 ( − i ∆ i T + ( − n +1 T ⊗ I where T ⊗ I is given by x T ( x ) ⊗ I . Lemma 8.1.2. (i) b = 0 , (ii) H is a bicomodule over itself with the right coaction ( id ⊗ ǫ )∆ . Proposition 8.1.3. The cohomology of the complex ( C, b ) is denoted by HH • ǫ ( H ) and one canshow that for n ≥ , HH n ǫ ( H ) is trivial. [5] It should be remarked that this definition of the coboundary operator strongly connected withthe Connes-Kreimer coproduct which yields the universality of the pair ( H CK , B + ) with respectto Hochschild cohomology theory. Indeed, b is defined in the sense that the grafting operator B + can be deduced as a Hochschild one cocycle. b T = 0 ⇐⇒ ∆( T ) = ( id ⊗ T )∆ + T ⊗ I . (8.1.1)Moreover, it is remarkable to see that these one cocycles determine generators of HH ǫ ( H ). Thusit is necessary to collect more information about Hochschild one cocycles. Lemma 8.1.4. (i) Let T be a generator of HH ǫ ( H ) , then T ( I ) is a primitive element of the Hopfalgebra.(ii) There is a surjective map HH ǫ ( H ) −→ P rim ( H ) , T T ( I ) .(iii) We can translate 1-cocycles to the universal enveloping algebra on the dual side and itmeans that for example 1-cocycle B + : H −→ H lin turns out to the dual map ( B + ) ⋆ : L −→ U ( L ) where that is a 1-cocycle in the Lie algebra cohomology. [5, 60] So it can be seen that primitive elements of the renormalization Hopf algebra and the graftingoperator B + can introduce an important part of the generators of HH ǫ ( H ). In fact, the sum overall primitive graphs (i.e. graphs without any divergent subgraphs which need renormalization)82f a given loop order n defines a 1-cocycle B + n such that every graph is generated in the rangeof these 1-cocycles. The importance of cocycles in the reformulation of Green functions will beobserved more clear, if we understand that every relevant tree or Feynman graph is in the rangeof a homogeneous Hochschild 1-cocycle of degree one.One should notice that insertion into a primitive graph commutes with the coproduct andtherefore it determines a generator of HH ǫ ( H CK (Φ)). Lemma 8.1.5. In a renormalizable theory Φ , for each r ∈ R + , we know that G rφ = φ (Γ r ) . Thereare Hochschild 1-cocycles B + k,r in their expansions such that they can be formulated with buildingblocks (i.e. 1PI primitives graphs) of the theory and it means that B + k,r = X γ ∈ H (Φ) (1) ∩ H lin (Φ) ,res ( γ )= r | sym ( γ ) | B + γ where the sum is over all Hopf algebra primitives γ contributing to the amplitude r at k loops.[34, 52, 59] In gauge theories, we may have some overlapping sub-divergences with different external struc-tures. Therefore it is possible to make a graph Γ by inserting one graph into another but in thecoproduct of Γ there may be subgraphs and cographs completely different from those which weused to make Γ. In this situation, it is impossible that each B +Γ (that Γ is 1PI) be a Hochschild1-cocycle. Since there may be graphs appearing on the right hand side of the formula in (8.1.1)which do not appear on the left hand side and for solving this problem in these theories, thereare identities between graphs. For example: Ward identities in QED and Slavnov-Taylor (ST)identities in QCD. These identities generate Hopf (co-)ideals in H CK ( QED ) and H CK ( QCD ),respectively where with working on the related quotient Hopf algebras, we will obtain Hochschild1-cocyles. [57, 58, 61, 67, 100, 101, 108] Lemma 8.1.6. The locality of counterterms and the finiteness of renormalized Green functionsare another applications of Hochschild cohomology theory in QFT. [6, 13, 54, 59] Another important influence of Hochschild cohomology in quantum field theory can be foundin rewriting quantum equations of motion in terms of Hopf algebra primitives and elements in H lin (Φ) ∩ { kerAug (2) /kerAug (1) } (8.1.2)such that it can be applied to characterize DSEs. Definition 8.1.7. For a given renormalizable theory Φ , let H be its associated free commutativeconnected graded Hopf algebra and ( B + γ n ) n ∈ N be a collection of Hochschild 1-cocycles (related tothe primitive 1PI graphs in H ). A class of combinatorial Dyson-Schwinger equations in H [[ α ]] has the form X = I + X n ≥ α n ω n B + γ n ( X n +1 ) such that ω n ∈ K and α is a constant. This family of equations provides a new source of Hopf subalgebras such that with working onrooted trees, some interesting relations between Hopf subalgebras of rooted trees (decorated byprimitive 1PI Feynman graphs) and Dyson-Schwinger equations (at the combinatorial level) canbe found. Theorem 8.1.1. Each combinatorial Dyson-Schwinger equation DSE has a unique solution (givenby c = ( c n ) n ∈ N , c n ∈ H ) such that it generates a Hopf subalgebra of H . [5, 6, 60] roof. The elements c n are determined by c = I ,c n = n X m =1 ω m B + γ m ( X k + ... + k m +1 = n − m,k i ≥ c k ...c k m +1 ) . And so the unique solution is given by X = P n ≥ α n c n . The related Hopf subalgebra structurefrom this unique solution is given by the following coproduct∆( c n ) = n X k =0 P nk ⊗ c k such that P nk := P l + ... + l k +1 = n − k c l ...c l k +1 are homogeneous polynomials of degree n − k in c l ( l ≥ n ). Remark 8.1.8. The independency of the coproduct from the scalars ω k determines an isomor-phism between the induced Hopf subalgebras by all DSEs of this class. It means that all Hopfsubalgebras which come from a fixed general class of DSEs are isomorphic. Lemma 8.1.9. For a connected graded Hopf algebra H and an element a ∈ H , set val ( a ) := max { n ∈ N : a ∈ M k ≥ n H k } . It defines a distance on H such that for each a, b ∈ H , d ( a, b ) := 2 − val ( a − b ) where its related topology is called n -adic topology. Lemma 8.1.10. The completion of H with this topology is denoted by H = Q ∞ n =0 H n such that itselements are written in the form P a n where a n ∈ H n . One can show that H has a Hopf algebrastructure originally based on H and in fact, the solution of a DSE belongs to this completion.[32, 33] Since Hopf algebra of Feynman diagrams of a given theory fundamentally defined by theinsertion of graphs into each other (namely, pre-Lie operation ⋆ ), therefore one can have a newdescription from Hochschild one cocycles in terms of this operator. In continue we consider thisnotion. Definition 8.1.11. For the Lie algebra g , let U ( g ) be its universal enveloping algebra. For a given g − module M (with the Lie algebra morphism g −→ End K ( M ) ), one can define a U ( g ) − bimodulestructure given by M ad = M with left and right U ( g ) − actions: X.m = X ( m ) , m.X = 0 . Definition 8.1.12. Suppose C nLie ( g , M ) := Hom ( V n g , M ) be the set of all alternating n − linearmaps f ( X , ..., X n ) on g with values in M . The Chevalley-Eilenberg complex is defined by M −→ δ C ( g , M ) −→ δ C ( g , M ) −→ ... such that δ ( f )( X , ..., X n +1 ) := X i Let H be the Hopf algebra of Feynman diagrams of the theory Φ and L : H −→ H lin a linear map with its dual L ⋆ : L −→ U ( L ) . Then ∆( L ) = ( id ⊗ L )∆ + L ⊗ I ⇐⇒ X Γ , Γ :1 P I ( n ( X , X , Γ ) − n ( X , X , Γ )) L ⋆ (Γ − Γ ) = X .L ⋆ ( X ) − X .L ⋆ ( X ) where Γ , Γ are 1PI graphs such that Γ /X = X , Γ /X = X . Corollary 8.1.14. For the Connes-Kreimer Hopf algebra of rooted trees, we have H nLie ( L CK , U ( L CK )) ≃ HH n ( U ( L CK ) , U ( L CK )) ≃ HH n ( H GL , H GL ) . Result 8.1.14 shows that when we want to identify one cocycles for the Hopf algebra H CK (atthe Lie algebra level), it is enough to find Hochschild one cocycles on the Hopf algebra H GL .It is possible to do above procedure for the universal Hopf algebra of renormalization to char-acterize its corresponding one cocycles at the Lie algebra level. It was shown that H U ≃ K [ T(A) ] I π such that A = ( f n ) n ∈ N . Therefore for each element u in K [ T(A) ] I π , there exist t ∈ H ( T ( A )) U and i ∈ I π such that u = t + i. (8.1.7)And furthermore there exist t , t , z , z , s ∈ T(A) such that u = t + t ◦ z + z ◦ t − t z + s ◦ ( t z ) + z ◦ ( t s ) + t ◦ ( sz ) − t z s. (8.1.8)85t is easy to see that u is a linear combination of rooted trees and therefore its coproduct shouldbe a linear combination of coproducts. It means that when each component of the above sum isprimitive, u is primitive. For example, coproduct of t ◦ z contains non-trivial terms that inducedby admissible cuts on rooted trees t , z and also admissible cuts on t ◦ z . If t ◦ z is primitive,then all of these terms should be canceled. Lemma 8.1.15. For a given primitive element u ∈ K [ T(A) ] I π , t , t , z , z , s are empty tree. Corollary 8.1.16. u = t + i ∈ K [ T(A) ] I π is primitive iff i be the empty tree and t primitive. With attention to the Hall set corresponding to H U and the related Lyndon words, a newcharacterization from primitives at this level can be given. Lemma 8.1.17. Let H ( T ( A )) U be the corresponding Hall set to the Lyndon words on the locallyfinite total order set A = ( f n ) n ∈ N . t ∈ H ( T ( A )) U is primitive iff t = B + a ( I ) for some a ∈ A . It means that H U has just primitive elements of degree zero (i.e. a vertex labeled by onearbitrary generator f n ) and its related one cocycle is denoted by B + f n . In fact, lemma 8.1.17is a representation of this note that a (Lyndon) word w is primitive if and only if w = a forsome a ∈ A . This story introduces generators (i.e. one cocycles) of HH ǫ ( H U ) correspondingto primitive elements of this Hopf algebra. On the other hand, theorem 8.1.2 makes clear theHochschild cohomology of H U at the Lie algebra level and it means that H nLie ( L U , U ( L U )) ∼ = HH n ( U ( L U ) , U ( L U )) . (8.1.9) Corollary 8.1.18. Lie polynomials are exactly the primitives in the graded dual of H U and there-fore for each Lie polynomial, one can identify its corresponding one cocycle at the level of the Liealgebra cohomology. Universal Hopf algebra of renormalization and factorization problem If we look at to the structure of the universal affine group scheme U and its corresponding Hopfalgebra H U , then it can be understood that this Hopf algebra is free from any physical dependency.In other words, it is independent of all renormalizable theories and this property separates thisspecific Hopf algebra from other renormalization Hopf algebras of Feynman diagrams of theories.In this part, with attention to the combinatorics of Dyson-Schwinger equations and on the basisof factorization problem of Feynman graphs, we are going to improve the independency of thespecific Hopf algebra H U to the level of non-perturbative study. This property leads us to removethe uniqueness problem of factorization and also, it plays an essential role to consider the behaviorof the Connes-Marcolli categorical approach with respect to these equations where as the conse-quence, we will explain in the next section that the Connes-Marcolli’s universal category E couldpreserve its universality at the level of Dyson-Schwinger equations.Kreimer discovered a very trickly interaction between Dyson-Schwinger equations and Eu-ler products and in this process he applied one important concept in physics namely, factoriza-tion. Factorization of Feynman diagrams into primitive components is in fact a brilliant highwayfor physicist to transfer information from perturbative theory (as a well-defined world) to non-perturbative theory (as an ill-defined world). This conceptional translation can be explained byDyson-Schwinger equations and we know that they are equations which formally solved in aninfinite series of graphs. On the other hand, it was shown that Feynman diagrams can be de-composed to primitive graphs with bidegree one in a recursive procedure such that the extensionof this mechanism to the level of DSEs is known as one important problem. Fortunately, withintroducing a new technical shuffle type product on 1PI Feynman graphs, one can find the answerof this question. One can rewrite the solution of a DSE based on this new shuffle type product.86he surprising note is that Euler factorization and Riemann ζ − function play a large role in thisprocess. [51, 52, 53, 54, 56, 59]With help of the given rooted tree reformulation of the universal Hopf algebra of renormaliza-tion and based on the combinatorial approach in the study of DSEs, we are going to consider thisfamily of equations at the level of H U and after that with respect to the uniqueness problem offactorization of Feynman diagrams, we shall see that how one can extend the universal propertyof this Hopf algebra to non-perturbative theory. This result can provide remarkable requirementsto find a new geometric interpretation from DSEs underlying a categorical configuration. [91, 96] Definition 8.2.1. For each labeled rooted tree t , the finite value w ( t ) := X v ∈ t [0] | dec ( v ) | is called decoration weight of t . Theorem 8.2.1. For a given equation DSE in the Hopf algebra H CK , there is an explicit presen-tation from the generators c n , ( n ∈ N ) (identified with the unique solution of DSE) at this level.We have c = I , c n = X t,w ( t )= n t | sym ( t ) | Y v ∈ t [0] ρ v such that ρ v = ω | dec ( v ) | ( | dec ( v ) | + 1)!( | dec ( v ) | + 1 − f er ( v ))! , for the case f ert ( v ) ≤ | dec ( v ) | + 1 and ρ v = 0 , for otherwise. [5] Corollary 7.2.27 allows us to lift this theorem to the level of H U . Proposition 8.2.2. For a given combinatorial equation DSE in H U , its unique solution c = ( c n ) n is determined by c = I π ,c n = X t ∈ H ( T ( A )) U ,t/ ∈ I π ,w ( t )= n t Y v ∈ t [0] ρ v + I π . Proof. It is proved by proposition 8.1.1, theorem 8.2.1 and this note that Hall trees (forests) haveno symmetries.Let K [[ h ]] be the ring of formal series in one variable over K . Foissy in [32, 33] considers someinteresting classes of DSEs (given by elements of K [[ h ]]) in Hopf algebras of rooted trees and also,he classifies systematically their associated Hopf subalgebras. One can lift his results to the levelof labeled rooted trees and hence H U . Definition 8.2.3. The composition of formal series gives a group structure on the set G := { h + X n ≥ a n h n +1 ∈ K [[ h ]] } . Hopf algebra of functions on the opposite of the group G is called Faa di Bruno Hopf algebra. It isa connected graded commutative non-cocommutative Hopf algebra and denoted by H F dB such thatfor each f ∈ H F dB and P, Q ∈ G , its coproduct is given by ∆( f )( P ⊗ Q ) = f ( Q ◦ P ) . emark 8.2.4. One can show that H F dB is the polynomial ring in variables Y i ( i ∈ N ), where Y i : G −→ K , h + X n ≥ a n h n +1 a i . Proposition 8.2.5. Let K [[ h ]] be the set of elements in K [[ h ]] with constant term and P ∈ K [[ h ]] .(i) Dyson-Schwinger equation X P = B + f n ( P ( X P )) in H U [[ h ]] has a unique solution with theassociated Hopf subalgebra H α,β U ( P ) iff there exists ( α, β ) ∈ K such that (1 − αβh ) P ′ ( h ) = αP ( h ) , P (0) = 1 (ii) For β = − , H ,β U ( P ) is isomorphic to the quotient Hopf algebra H FdB I π .(iii) H , − U ( P ) is isomorphic to a quotient Hopf algebra SY MJ of Hopf algebra of symmetricfunctions.(iv) H , U ( P ) is isomorphic to the quotient Hopf algebra K [ • ] I π .Proof. It is proved based on the rooted tree version of H U and the given results in sections 3, 4, 5in [32]. For the third part, there is a homomorphism θ : SY M −→ H CK that sends each generator m (1 , ..., | {z } n to the ladder tree l n . Set J := θ − ( I π ).Moreover, Hoffman suggests a new procedure to study DSEs with translating equations toa quotient of noncommutative version of the Connes-Kreimer Hopf algebra namely, Foissy Hopfalgebra. One can improve his main result to our interesting level. Proposition 8.2.6. The unique solution of the equation X = I π + B + f n ( X p ) in H U where p ∈ R is determined by t n = X t ∈ H ( T ( A )) U e ( t ) C p ( t ) t such that e ( t ) is the number of Hall planar rooted trees s such that α ( s ) = t (defined in theorem3.2.3).Proof. The unique solution of DSE is given by a formal sum X = I π + t + t + ... such that t n is a Hall tree in H ( T ( A )) U with degree n . Set e X := t + t + ... Equation DSE can be changed to the form e X = B + f n (( I π + e X ) p ) . Since the operator B + f n increases degree, it is easy to see that t n +1 = B + f n ( { I π + ( p e X + ( p e X + ... } n )88uch that {} n is the component of degree n . There is a natural homomorphism α : H F −→ H CK that maps each planar rooted tree to its corresponding rooted tree without notice to the orderin products. One can lift this homomorphism to the level of labeled rooted trees and apply it tostudy the given DSE at the level of the quotient Hopf algebra H F ( A ) I π . Let I π + e Y = I π + s + s + ... be its solution in this new level such that s n is a Hall planar rooted tree of degree n in H ( T ( A )) U .Induction shows that s n = X t ∈ H ( P n − ( A )) U C p ( t ) t such that C p ( t ) = Y v ∈ V ( t ) (cid:0) pc ( v ))where c ( v ) is the number of leaves of v , V ( t ) is the set of vertices of t with c ( v ) = 0 and H ( P n − ( A )) U is the Hall subset of H ( T ( A )) U generated by planar rooted trees of degree n .It is observed that for each Hall planar rooted tree s , C p ( s ) = C p ( α ( s )). Since α ( e Y ) = e X ,according to the theorem 6.2 in [41], one can obtain a clear presentation from the unique answerof DSE.Now it is the time to consider the attractive relation between Dyson-Schwinger equations andEuler products underlying the factorization problem. Definition 8.2.7. The analytic continuation of the sum P n n s is called Riemann ζ − function. If R ( s ) > , then it has an Euler product over all prime numbers given by ζ ( s ) = Q p − p − s . Definition 8.2.8. For a sequence of primitive 1PI graphs J = ( γ , ..., γ k ) in the renormalizabletheory Φ , a Feynman graph Γ is called compatible with J (i.e. Γ ∼ J ), if < Z γ ⊗ ... ⊗ Z γ k , ∆ k − (Γ) > = 1 . Lemma 8.2.9. Let n Γ be the number of compatible sequences with Γ . Define a product on 1PIgraphs given by Γ ⊎ Γ := X I ∼ Γ ,I ∼ Γ X Γ ∼ I ⋆I n Γ Γ . It is a commutative associative product. Lemma 8.2.10. Define a relation Γ ≤ Γ ⇐⇒ < Z − Γ , Γ > = 0 . It determines a partial order relation based on subgraphs. Remark 8.2.11. (i) One can rewrite the Connes-Kreimer coproduct by ∆(Γ) = X Γ , Γ ζ (Γ , Γ )Γ ⊗ Γ where if Γ ≤ Γ , then ζ (Γ , Γ ) = 1 and otherwise ζ (Γ , Γ ) = 0 .(ii) The product ⊎ is a generalization of the shuffle product ⋆ and it means that ⋆ appropriatesfor totally ordered sequences whenever ⊎ is just for partial order relation (on subgraphs). [51, 52] Lemma 8.2.12. Unique solution of the equation X = I + X γ α k γ B + γ ( X k γ ) (such that k γ is the degree of γ ) has an ⊎− Euler product given by X = ⊎ Y γ − α k γ γ . [51, 52, 54] Remark 8.2.13. The uniqueness of this factorization in gauge theories is lost and for removalthis problem, we should work on the quotient Hopf algebras. [52, 54, 99, 100, 101] Now we know that uniqueness is the main problem of factorization and in continue we wantto focus on this lack to show the advantage of the universal Hopf algebra of renormalization. Theorem 8.2.2. Consider set H pr of all sequences ( p , ..., p k ) of prime numbers such that theempty sequence is denoted by and define a map B + p such that its application on a sequence J = ( p , ..., p k ) is the new sequence ( p, p , ..., p k ) . There is a Hopf algebra structure on H pr suchthat its compatible coproduct with the shuffle product is determined by ∆( B + p ( J )) = B + p ( J ) ⊗ id ⊗ B + p ]∆( J ) , ∆(1) = 1 ⊗ , ∆(( p )) = ( p ) ⊗ ⊗ ( p ) .B + p ( J ) ⋆ B + p ( J ) = B + p ( J ⋆ B + p ( J )) + B + p ( B + p ( J ) ⋆ J ) . [57] Remark 8.2.14. It is natural to think that the operator B + p is a Hochschild one cocycle and itmeans that with this operator one can introduce Dyson-Schwinger equations at the level of H pr . Consider Dyson-Schwinger type equation X ( α ) = 1 + X p αB + p [ X ( α )] (8.2.1)in H pr . It has a decomposition given by X ( α ) = ⋆ Y p − α ( p ) . (8.2.2)Because of the shuffle nature of this Hopf algebra, one can suggest to apply universal Hopfalgebra of renormalization. It is observed that for finding the Euler factorization, one shoulddefine a new product on Feynman diagrams of a theory and on the other hand, the uniquenessof this factorization is not available in general. Now we see that with working at the level of H U ,this indicated problem can be solved. Proposition 8.2.15. There exists an Euler factorization in the universal Hopf algebra of renor-malization. roof. There are two shuffle structures to apply in H U for receiving factorization. One of themnamely, ⋆ is exactly the same as the shuffle product on H U . One can show that(i) The product of H U is integral,(ii) There is a combinatorial ⋆ − Euler product (comes from a class of DSEs) in the universalHopf algebra of renormalization.For the second claim, it is clear that for each variable f n ( n ∈ N > ) in H U , B + f n is a Hochschildone cocycle. Consider equation X ( α ) = 1 + X f n αB + f n [ X ( α )]in H U . By replacing prime numbers with these variables in (8.2.2) and also with notice to theshuffle structure in H U , one can get the decomposition X ( α ) = ⋆ Y f n − α ( f n )such that ( f n ) is a word with length one.It was shown that an extension of the shuffle product can be applied to obtain a factorizationfor the formal series of Feynman diagrams of the solution of a DSE. For the word w = f k ...f k n in H U , a word v is called compatible with w (i.e. v ∼ w ), if < p f k ⊗ ... ⊗ p f kn , ∆ k n − ( v ) > = 1 (8.2.3)such that p f n ≡ e − n s are Hall polynomials of primitive elements f n s. (One should stress that Hallpolynomials form a basis (at the vector space level) for the Lie algebra). Therefore v ∼ w ⇐⇒ < e − k ⊗ ... ⊗ e − k n , ∆ k n − ( v ) > = 1 . (8.2.4)Let n w be the number of compatible words with w . It determines our interesting product w ⊎ w := X v ∼ w ,v ∼ w X w ∼ v ⋆v n w w. (8.2.5)Now consider Dyson-Schwinger equation X = 1 + X f n α n B + f n ( X n ) (8.2.6)in the universal Hopf algebra of renormalization. Its unique solution has an ⊎− Euler productgiven by X = ⊎ Y f n − α n f n . (8.2.7) Corollary 8.2.16. There is a unique factorization into the Euler product in the universal Hopfalgebra of renormalization.Proof. With attention to the given coproduct in theorem 7.1.1 and also proposition 8.2.15, it canbe seen that v ∼ w ⇐⇒ v = w = ⇒ n w = 1 = ⇒ w ⊎ w = w ⋆ w . Therefore ⊎− Euler product and ⋆ − Euler product in the universal Hopf algebra of renormalizationare the same. On the other hand, we know that each word has a unique representation with adecreasing decomposition to Hall polynomials and moreover, corollary 7.2.45 shows that theseelements determine a basis at the vector space level for the free algebra H U .91t last, let us consider the possibility of defining the Riemann ζ − function in H U . Proposition 8.2.17. The Riemann ζ − function can be reproduced from a class of DSEs in theuniversal Hopf algebra of renormalization.Proof. One can define an injective homomorphism from H prime to H U given by J = ( p , ..., p n ) w J = f p ...f p n . There is an interesting notion (in [51]) to obtain Riemann ζ − function from a class of DSEs in H prime such that one can lift it to the level of H U . Consider the equation X ( α ) = 1 + X p αB + f p [ X ( α )]in H U such that the sum is on prime numbers. Choose a homomorphism φ s of H U given by φ s ( w ) = 1 | w | ! pr ( w ) − s such that for the word w = f k ...f k n , pr ( w ) := k ...k n . It is observed that lim α −→ φ s [ X ( α )] = ζ ( s ) . The Euler product of the above DSE is given by X ( α ) = ⋆ Y p − α ( f p )and one can see that φ s ( ⋆ Y p − α ( f p ) ) = Y p − p − s = ζ ( s ) . Roughly speaking, this explained procedure yields one essential reason to generalize the conceptof universality of H U in non-perturbative theory. Corollary 8.2.18. Universal Hopf algebra of renormalization can preserve its independency fromphysical theories at the level of Dyson-Schwinger equations and therefore non-perturbative theory. Categorical configuration in the study of DSEs We discussed that how combinatorial Dyson-Schwinger equations can determine a systematicformalism to consider non-perturbative theory based on the Connes-Kreimer perturbative renor-malization. It means that this Hopf algebraic reinterpretation can lead to an extremely practicalstrategy to discover some unknown parts of the theory of quantum fields. In this process we un-derstood that from each DSE one can associate a Hopf subalgebra of Feynman diagrams of a fixedtheory. In this part we want to work on these Hopf subalgebras and introduce a new frameworkto study Dyson-Schwinger equations based on the Connes-Marcolli’s universal approach. Thispurpose can be derived by finding an interrelationship between these equations and objects ofthe category of flat equi-singular vector bundles where it has an essential universal property inthe mathematical treatment of the perturbative renormalization. In this generalization one cancharacterize a new family of equations (i.e. universal Dyson-Schwinger equations ) such that it92stablishes a new procedure to calculate Feynman integrals in the sense that at first, one can findthe solution of an equation at the simplified universal level (i.e. DSEs in H U ) and then with usinggraded representations (introduced by Connes-Marcolli theory), we will project this solution tothe level of an arbitrary renormalizable theory. In addition, by this way, one can find an arterialroad from combinatorial DSEs to the universal category E where as the consequence, it mentionsa new geometric interpretation from Dyson-Schwinger equations underlying the Riemann-Hilbertcorrespondence. So it makes possible to expand the universality of the category E to the level ofthese equations. [91, 96]The universality of the category E is determined by its relationship with categories connectedwith renormalizable theories and it was shown in theorem 6.2.2 that this neutral Tannakian cate-gory enables to cover categories of all renormalizable theories as full subcategories and moreover,it determines the specific affine group scheme U ∗ . Here we are going to apply Hopf subalgebrastructures depended on Dyson-Schwinger equations to find a relation between these equations andobjects of E .Let us consider the Dyson-Schwinger equation DSE in H [[ α ]] with the associated Hopf subal-gebra H c and affine group scheme G c and let g c be its corresponding Lie algebra. Lemma 8.3.1. Generators of the Lie algebra g c are linear maps Z n : H c −→ C , Z n ( c l ) = δ n,l . Lemma 8.3.2. One can make a trivial principal G c − bundle P c = B × G c over the base space B = ∆ × G m ( C ) such that its restriction on B is denoted by P c = B × G c . Corollary 8.3.3. (i) Equivalence relation given by definition 6.1.3 provides a bijective correspon-dence between minus parts of the Birkhoff decomposition of loops (with values in G c ) and elementsof the Lie algebra g c .(ii) Theorem 6.1.3 guarantees the existence of the classes of equi-singular flat connectionswith respect to the elements of this Lie algebra. Therefore a classification of equi-singular flatconnections on the vector bundle connected with the combinatorial equation DSE in the theory Φ will be determined. Corollary 8.3.4. For a given equation DSE in the theory Φ , equivalence classes of flat equi-singular G c − connections on P c are represented by elements of the Lie algebra g c and also, eachelement of this Lie algebra identifies one specific class of equi-singular G c − connections. Thisprocess is done independent of the choice of a local regular section σ : ∆ −→ B with σ (0) = y . Proposition 8.3.5. Let c = ( c n ) n ∈ N be the unique solution of DSE. Result 8.3.4 shows that foreach k ∈ N there exists a unique class of flat equi-singular connections ω kc on P c such that ω kc ∼ D γ Z k ,γ Z k ( z, v ) = T e − z R v u Y ( Z k ) duu , u = tv, t ∈ [0 , . Let V l be an arbitrary l − dimensional vector space generated by some elements of ( c n ) n ∈ N and ψ lc : G c −→ Gl ( V l ) be a graded representation. By theorem 6.2.3, the pair ( ω kc , ψ lc ) identifies anelement from the category of flat equi-singular vector bundles E . Corollary 8.3.6. For a given Dyson-Schwinger equation DSE in the theory Φ , a family of objectsof the category E will be determined. When we consider the Riemann-Hilbert correspondence underlying the Connes-Kreimer theoryof perturbative renormalization, the universality of the category E is characterized by this factthat E carries a geometric representation from all renormalizable theories as subcategories. Nowthis notion can be expanded to the level of combinatorial DSEs.93 efinition 8.3.7. For a fixed equation DSE in the theory Φ , one can define a subcategory E Φ c of E such that its objects are introduced by corollary 8.3.6. Proposition 8.3.8. For each given equation DSE, there are classes of flat equi-singular G c -connections such that they introduce a category. Instead of working on this category, one can go toa universal framework and concentrate on a full subcategory of E of those flat equi-singular vectorbundles that are equivalent to the finite dimensional linear representations of G ∗ c . It provides thisfact that E Φ c has power to store a geometric description from DSE. With attention to the algebro-geometric dictionary (in the minimal subtraction scheme indimensional regularization [16]), we know that each loop γ µ in the space of diffeographisms canassociate a homomorphism φ γ µ : H −→ C and then we can perform Birkhoff decomposition atthe level of these homomorphisms to obtain physical information. Lemma 8.3.9. There is a surjective map from the affine group scheme G to the affine groupscheme G c .Proof. We know that H c is a Hopf subalgebra of H . With restriction one can map each element φ in the complex Lie group scheme G ( C ) to its corresponding element φ c ∈ G c ( C ). On the otherhand, there is an injection from H c to H such that it determines an epimorphism from Spec ( H )to Spec ( H c ). Proposition 8.3.10. Objects of the category E Φ c store some parts of physical information of thetheory with respect to the Dyson-Schwinger equation DSE. This fact shows that with help the of objects of the category of flat equi-singular vector bundles,a geometrically analysis from all of the combinatorial DSEs in a given theory can be investigatedand since according to [57] these equations address non-perturbative circumstances, thereforecategory E can preserve its universal property at this new level.It was shown that Hopf algebras of renormalizable physical theories and H U have the samecombinatorial source (namely, rooted trees). By applying the next fact, one can find an interestingidea to compare Dyson-Schwinger equations at the level of renormalizable physical theories withtheir corresponding at the level of the universal Hopf algebra of renormalization.We know that with the help of Hochschild one cocycles (identified by primitive elements ofthe Hopf algebra), one can characterize combinatorial DSEs. Fix an equation DSE in H withthe associated Hopf subalgebra H c . According to theorem 6.2.3, for the equivalence class of flatequi-singular connections ω on e P one can identify a graded representation ρ ω . Let ω c be theflat equi-singular connection on e P c := B × G ∗ c corresponding to ω with the associated gradedrepresentation ρ ω c . On the other hand, one can consider DSE in a decorated version of theConnes-Kreimer Hopf algebra of rooted trees. Theorem 8.1.1 provides an explicit reformulationfrom generators of the Hopf algebra H c such that by proposition 8.2.2, we can lift these generatorsto the level of the rooted tree representation of H U . This process determines a new equation DSE u and a Hopf subalgebra H u of H U with the related affine group scheme U c . Lemma 8.3.11. For a fixed equi-singular flat connection ω , one can provide a graded represen-tation ρ cω c : U ∗ c −→ G ∗ c such that it is a lift of the representation ρ ω and characterized with ρ ω c .In summary, we have the following commutative diagram. U ∗ (cid:29) (cid:29) ρ ω ρ ωc (cid:24) (cid:24) G ∗ c O O ρ cωc o o G ∗ U ∗ c (8.3.1)94 roposition 8.3.12. The morphism ρ cω c induced by representations ρ ω , ρ ω c provides the conceptof universal Dyson-Schwinger equations and it means that DSE u maps to DSE (or DSE lifts to DSE u ) under the representation ρ cω c . When K = Q , the fiber functor ϕ : E Q −→ V Q is given by ϕ = L ϕ n such that for each elementΘ of the category, ϕ n (Θ) := Hom ( Q ( n ) , Gr W − n (Θ)) . (8.3.2)For each n , Q ( n ) = [ V, ▽ ] is an object in E Q such that V is an one dimensional Z − graded Q − vectorspace placed in degree n and ▽ = d (i.e. ordinary differentiation in one variable). We explainedthat how one can identify some elements of this category with objects given in corollary 8.3.6 andnow it would be remarkable to see that elements Q ( n ) are represented with respect to a given DSEand for this work, the class of trivial connections should be calculated. Because proposition 1.101in [19] provides this fact that the connection ▽ identifies a connection ω where ▽ = d + ω . In otherwords, we have to find an element in the Lie algebra g c such that its corresponding equi-singularconnection is equivalent to 0 and one can show that Z plays this role. We have θ − t ( Z k ) = e − tk Z k (8.3.3) k = 0 = ⇒ θ − t ( Z ) = Z : H c −→ C , Z ( c k ) = δ ,k . (8.3.4)By theorem 1.60 in [19], for the element β = Z in g c , we have γ c − ( z ) = T e − Z z R ∞ dt . (8.3.5)On the other hand, we know that the expression T e R u α ( t ) dt , such that α ( t ) = 1 for t = t and α ( t ) = 0 for otherwise, is the value g ( u ) of the solution for the equation dg ( u ) = g ( u ) α ( u ) du, g (0) = 1 . (8.3.6)Therefore by (8.3.6), it can be seen that g ( t ) = const. = ⇒ D g = 0 = ⇒ ω ∼ . (8.3.7) Corollary 8.3.13. One can represent elements Q ( n ) with a given equation DSE.Proof. Let c = ( c n ) n ∈ N be the unique solution of a fixed equation DSE and V be the one di-mensional Q − vector space generated by c n placed in degree n . It is observed that ω ∼ Dγ cZ such that γ cZ is the constant loop and it means that ω = 0. Since ▽ = d + ω , the proof iscomplete.One should note that there are different choices to define the vector space V in the representa-tion of Q ( n ) with a fixed DSE but all of them belong to the isomorphism class of one dimensional Z − graded Q − vector spaces. For a given equation DSE, result 8.3.13 shows that the equation(8.3.2) can play the role of the fiber functor for the full abelian tensor category E Φ c .95 hapter 9 Conclusion and future improvements Here we try to have an overview from the whole structure of this work and then with attentionto the given results, we will introduce some major problems which help to advance theory ofintegrable systems and theoretical understanding of non-perturbative phenomena underlying theConnes-Kreimer-Marcolli postulates. Overview Roughly speaking, this research attempts to establish new progresses in the study of quantum fieldtheory underlying renormalization Hopf algebra and in particular, we focused on three importantproblems namely, theory of quantum integrable systems, combinatorial representations of universalHopf algebra and universal singular frame and categorification of non-perturbative studies (withrespect to the Kreimer’s approach).The first purpose was focused on integrable systems. Firstly, with the help of non-commutativedifferential forms, we considered an essential problem in quantum field theory namely, quantumintegrable systems. The key point in our chosen approach is summarized in the deformation ofalgebras based on Nijenhuis operators where these maps are induced from regularization or renor-malization schemes. Indeed, multiplicativity of renormalization reports about a hidden algebraicnature inside of the BPHZ method namely, Rota-Baxter structure such that on the basis of thisproperty we could introduce a new family of quantum Hamiltonian systems depended on renor-malization or regularization schemes. Secondly, we studied motion integrals related to Feynmanrules characters and then it was observed that how Connes-Kreimer renormalization group makespossible to obtain an infinite dimensional integrable quantum Hamiltonian system. Thirdly, basedon Bogoliubov character and BCH formula, we found a new class of fixed point equations re-lated to motion integrals where it yields to search more geometrical meanings inside of physicalparameters.The second purpose was concentrated on very special object in the Connes-Marcolli universalinterpretation of renormalizable quantum field theories. We analyzed universal Hopf algebra ofrenormalization and also with the help of its shuffle nature, we reproduced it with Hall rooted trees.This new description provided a strong tool to consider the relation between H U and some othercombinatorial Hopf algebras. Moreover, we developed this rooted tree reformulation to the levelof the universal affine group scheme and its related Lie algebra. From this story, we investigatedthat how theory of operads and poset theory (i.e. Hall rooted trees and Hall polynomials) can beentered into the study of quantum field theory. According to the universal property of H U amongall renormalizable physical theories and based on its large role in finding an isomorphism betweenuniversal category of flat equi-singular vector bundles and the category of finite dimensional linearrepresentations of universal affine group scheme U ∗ , it was explained that how we can apply this96ooted tree version to obtain a Hall tree-Hall polynomial representation from universal singularframe.The third purpose can be summarized in working on a geometric studying of non-perturbativetheory. This project was done in two steps such that in the first stage, the extension of indepen-dency of the universal Hopf algebra of renormalization to the level of Dyson-Schwinger equationswas considered where in this process, we studied DSEs at the level of H U . Furthermore, withattention to factorization of Feynman diagrams (determined with solutions of Dyson-Schwigerequations), we showed that this factorization uniquely exists in H U and also, one can reformulatethe Riemann ζ − function with DSEs at the level of this particular Hopf algebra. In the next stage,with the help of generated Hopf subalgebras by DSEs, we showed that the category of equi-singularflat bundles can recover all of these equations. Generalization of the categorical interpretation (inthe study of renormalizable theories) to Kreimer’s non-perturbative modeling and introducing theconcept of universal Dyson-Schwinger equations were immediate observations in this direction. Other integrable systems Renormalization group determined the compatibility of the introduced integrable Hamlitoniansystems with the Connes-Kreimer perturbative renormalization. It should be interesting to findanother integrable Hamiltonian systems. For instance with deforming the algebra L ( H, A ) un-derlying Nijenhuis maps (determined with regularization schemes), one can have this chance. Onthe other hand, finding a physical theory such that components of Birkhoff factorization of itsassociated Feynman rules character play the role of motion integrals for the character, can be veryimportant question. We want to know that is there any specific algebro-geometric property in thiskind of theories? Relation between motion integrals and DSEs Working on this question can help us to develop our approach to integrable systems to the level ofnon-perturbative theory and further, it leads to discover a new description from quantum motionsbased on Nijenhuis type Poisson brackets. It seems that with this notion one can introduce a newinterrelationship between theory of Rota-Baxter algebras and Dyson-Schwinger equations. 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Yeats, Growth estimates for Dyson-Schwinger equations , Ph.D. thesis, arXiv:0810.2249v1, 2008. ndex A Admissible cut 12, 13, 72, 86Alphabetical order 69Amplitude 1, 23, 26, 27, 28, 29, 82, 83Antipode 7, 8, 13, 33, 65, 72Augmentation ideal 9, 24, 25, 29Augmented modified quasi-shuffle product 39Augmented operad 17, 75 B Balanced bracket representation 14, 15, 72BCH series 53, 54, 55, 80, 96Beta function 2, 10, 29, 32, 49, 53, 56, 61, 80Bialgebra 6, 7, 8, 51Birkhoff decomposition 2, 13, 20, 29, 30, 31, 32, 33, 34, 49, 57, 58, 59, 61, 80, 93, 94Bogoliubov 1, 28, 31, 33, 34, 53, 55, 80, 96BPHZ 1, 2, 12, 20, 28, 31, 32, 33, 47, 52, 53, 57, 59, 96 C Casimir function 52Coinvariant 75Connected filtered 8Connected graded 7, 9, 10, 13, 14, 15, 16, 23, 28, 57, 61, 62, 64, 69, 74, 83, 84, 87Connes-Kreimer Hopf algebra 3, 11, 13, 14, 17, 19, 23, 24, 25, 28, 33, 40, 53, 55, 62, 63, 70, 75, 85, 88,94Connes-Marcolli 3, 10, 19, 57, 61, 62, 66, 77, 81, 86, 92, 93, 96Connes-Moscovici Hopf algebra 15Conservation law 21Convolution product 6, 8, 13, 29, 33, 54Counterterm 1, 2, 3, 10, 20, 24, 28, 29, 31, 33, 34, 57, 59, 61, 62, 77, 80, 83 D deRham complex 42Deformed algebra 40, 51, 54Dimensional Regularization 1, 2, 3, 20, 29, 30, 32, 33, 34, 47, 48, 57, 59, 62, 94Diffeographism 31, 32Dimensionally regularized Feynman rules character 2, 31, 33, 47, 48, 49, 50, 51, 52, 80Dyson-Schwinger equation 3, 15, 24, 62, 81, 83, 86, 88, 89, 90, 91, 92, 93, 94, 95, 97 Elimination 1, 13Equi-singular 2, 3, 4, 33, 57, 58, 59, 60, 61, 62, 77, 82, 92, 93, 94, 95, 96, 97External edge 21, 22, 26, 27, 28, 83 F Faa di Bruno Hopf algebra 87Feynman diagram 1, 2, 3, 4, 8, 10, 11, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 37, 40, 48, 49, 51,63, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 97Fibre functor 60, 61, 95 G Gluon 3Graded representation 61, 62, 80, 93, 94Grafting 11, 19, 25, 66, 82Green function 26, 27, 83Grossman-Larson Hopf algebra 14 H Hall forest 67, 68, 71, 77, 80Hall polynomial 63, 76, 77, 78, 79, 80, 91, 96, 97Hall set 67, 68, 69, 71, 76, 77, 80, 86Hamiltonian derivation (vector field) 36, 40, 42, 43Hochschild one cocycle 11, 14, 27, 81, 82, 83, 84, 85, 90, 91, 94Hoffman pairing 64 I Idempotent 31, 38, 40, 41, 43, 44, 46, 48, 49, 50, 53Incidence Hopf algebra 17, 18, 19, 71, 75Infinitesimal generator 2, 10, 29, 32, 33, 34, 56, 80Insertion 13, 21, 23, 27, 83, 84Integral of motion (motion integral) 34, 36, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 96, 97Integrable system 2, 3, 30, 33, 34, 35, 36, 37, 39, 40, 45, 46, 50, 51, 52, 56, 96, 97Internal edge 8, 21, 22, 23, 28 J Jacobi identity 37, 42, 43 L Ladder tree 13, 16, 17, 53, 54, 72, 73, 74, 88Leibniz 41, 43Letter 38, 64, 69, 76Lie algebra cohomology 82, 84, 85, 86Lie polynomial 65, 66, 76, 86Logarithmic 9, 27, 58Loop algebra 52Lyndon word 17, 69, 71, 73, 78, 86