Modified graded Hennings invariants from unrolled quantum groups and modified integral
MMODIFIED GRADED HENNINGS INVARIANTS FROMUNROLLED QUANTUM GROUPS AND MODIFIEDINTEGRAL
NATHAN GEER, NGOC PHU HA, AND BERTRAND PATUREAU-MIRAND
Abstract.
The second author constructed a topological ribbonHopf algebra from the unrolled quantum group associated with thesuper Lie algebra sl (2 | MSC: 57M27, 17B37.Key words: Unrolled quantum group, topological ribbon Hopf al-gebra, Hennings type invariant, discrete Fourier transform, modifiedintegral. 1.
Introduction
It is known that the category of modules over the semi-restrictedquantum group at a root of unity produces 3-manifold invariants (see[6]). In [17], the second author used the semi-restricted quantum groupassociated to the super Lie algebra sl (2 |
1) (not its category of modules)to define a Virelizier-Hennings type 3-manifold invariant (see [20, 35]).Here we generalize the latter construction to a large setting of Hopfalgebras.For the simplest example of a semi-restricted quantum group see Ex-ample 2.3 below. By adding Cartan elements this quantum group hasbeen extended to a Hopf algebra called the unrolled quantum group,see for example [6, 12, 7]. In [1], Andruskiewitsch and Schweigert con-sider unrolled Hopf algebras which are a generalization of the unrolledquantum groups containing the case of Lie (super)algebras and moregeneral diagonal Nichols algebra (see [19]). In this paper we propose ageneral construction which should produce quantum 3-manifold invari-ants for all of these unrolled Hopf algebras. To prove this one needs toshow these algebras satisfy the five axioms listed in this paper.Let us now summarize these axioms. Starting with a free abeliangroup Λ of rank r we consider three algebras U , U H and U = U /I a r X i v : . [ m a t h . QA ] J un NATHAN GEER, NGOC PHU HA, AND BERTRAND PATUREAU-MIRAND which are generalizations of the semi-restricted, unrolled and smallquantum groups, respectively (for the simplest example of these quan-tum groups see Example 2.3 below). Loosely speaking, the axioms werequire are:Axiom 1: U is a Λ-graded pivotal Hopf algebra (graded by “weights”).Axiom 2: There exists a quasi R-matrix for U H .Axiom 3: U is unimodular.Axiom 4: The integral of the twist and of its inverse are non zero.Axiom 5: There exists a projective U H -module whose restriction to a U -module remains projective.We will prove that the semi-restricted, unrolled and small quantumgroups associated to a simple Lie algebra of rank r satisfy these fiveaxioms and thus lead to the 3-manifold invariants define in this paper.The first main result of this paper is to embed the unrolled quan-tum group U H into a topological ribbon Hopf algebra (cid:99) U H which has atopology of a complete nuclear space. In particular, we give a comple-tion of the unrolled quantum group which is a topological ribbon Hopfalgebra describe as follows. The subalgebra generated by the CartanLie algebra H in this unrolled quantum group is embedded into thespace C ω ( H ∗ ) of holomorphic functions on the dual space H ∗ . Thenthe unrolled quantum group is embedded in its completion which is acomplete nuclear space and has the topology of uniform convergenceon compact sets. Using this completion we show the quasitriangularstructure of the small quantum group can be lifted to a topologicalquasitriangular structure and a topological ribbon Hopf algebra. Thisleads to a large class of topological ribbon Hopf algebra. In a differ-ent context, Markus J. Pflauma and Martin Schottenloher [28] alreadyconsider other kinds of nuclear Hopf algebras and their holomorphicdeformations.The techniques discussed above were first used by the second authorto defined a topological ribbon Hopf algebra from the super Lie algebra sl (2 | G -integral to con-struct an invariant of Virelizier-Hennings type of 3-manifolds decoratedby a cohomology class.Using different techniques a modifed Hennings type invariant is givenin [10]. This is an invariant defined on pairs ( M, T ) consisting of a 3-manifold M and a bichrome graph T inside. The main ingredients ofthe construction in [10] consists of a finite dimensional Hopf algebra H and a modified trace of the category H -mod of finite dimensionalleft modules over H . The second main result of this paper is to definea modified graded Hennings type invariant for the unrolled quantumgroups satisfying the five axioms of this paper. ODIFIED GRADED HENNINGS INVARIANT 3
The modified invariants of this paper are thus graded versions ofthe modified Hennings invariants of [10]. In [9] the later invariants areshown to give a (non-graded) Hennings type formula for the Reshetikhin-Turaev type quantum invariants of [6] associated to zero cohomologyclasses. Here we give a similar modified graded Hennings-Vireliziertype formula for the graded version of the invariants of [6] with non-zero cohomology classes. To do this we introduce a new algebraic toolcalled the modified integral.Let us discuss the organization of the paper. In Section 2, we recallsome needed properties of unrolled quantum groups and their repre-sentations. Section 3 contains the construction of a topological ribbonHopf algebra from an unrolled quantum group. It also describes somespecial elements called power elements which will be used to color themixed coupons of the bichrome graphs. In Section 5 we adapt the con-struction of the universal invariant associated to a ribbon Hopf algebra.In Section 6 we define both the graded Hennings invariant H and itsmodified version H (cid:48) which are invariants of a admissible compatibletriple ( M, Γ , ω ) where M is a closed 3-manifold, Γ is a bichrome graphinside M and ω is a cohomology class of H ( M \ Γ; G ). In Section 7, wepresent the notion of a modified integral which allows us to relax theadmissibility condition for the modified graded Hennings invariant andprovides another way to determine the invariant. Finally, in Section 8we explain how this invariant generalize some previously defined nonsemi-simple invariants. Acknowledgments.
Nathan Geer was partially supported by the NSFgrant DMS-1452093. This grant also supported Ngoc Phu Ha to visitUtah State University where much of the work of this paper took place.2.
Quasi-triangular unrolled Hopf Algebras
In this section we give our definition of a quasi-triangular unrolledHopf Algebra and consider its category of weight modules. Similaralgebras have been considered in [1]. We discuss why the previouslydefined unrolled quantum groups of [6, 12, 7, 13] are examples of thealgebras define in this section.2.1.
Unrolled Hopf Algebras.
Fix an integer (cid:96) and a (cid:96) th root ofunity ξ = e iπ(cid:96) . If ( G, +) is a abelian group and V is a vector spacewe say V is G -graded if there is a decomposition V = (cid:76) g ∈ G V g whereeach V g is a vector space. We say an algebra V is G -graded if it itsunderlying vector space is G -graded, 1 ∈ G and the multiplicationpreserves the grading: if v ∈ V g and w ∈ V h then vw ∈ V g + h . If v ∈ V g we say v is homogeneous .Let Λ be a free abelian group of rank r and W be a Λ-graded finitedimensional vector space over C with a special degree 0 element denoted1 W . If w ∈ W is homogeneous, then we denote its degree by | w | ∈ Λ. NATHAN GEER, NGOC PHU HA, AND BERTRAND PATUREAU-MIRAND
Let Λ ∗ = Hom Z (Λ , Z ) be the abelian group of group morphism betweenΛ and Z . Fix a basis { a , . . . , a r } of Λ ∗ . Finally, let H = Λ ∗ ⊗ Z C .The group ring C [Λ ∗ ] is the free vector space on Λ ∗ over C whichis generated by the formal variables { K a : a ∈ Λ ∗ } , with the relation K a + b = K a K b for any a, b ∈ Λ ∗ . In particular, C [Λ ∗ ] can be identifiedwith the ring of Laurent polynomials in the r variables K i := K a i .It has the structure of a Hopf algebra where each element of the set { K a : a ∈ Λ ∗ } is a group-like element.Recall that a Hopf algebra is pivotal if the square of the antipodecan be expressed via the conjugation by a group-like element, calledpivot g . We assume the following axiom. Axiom 1.
There exists a Λ -graded pivotal Hopf algebra U with under-lying vector space U = W ⊗ C C [Λ ∗ ] which is an extension of the Hopf algebra C [Λ ∗ ] = 1 W ⊗ C [Λ ∗ ] and inwhich for any homogenous x ∈ W ∼ = W ⊗ the following relation holds: (1) K a xK − a = ξ a ( | x | ) x for all a ∈ Λ ∗ . We denote the comultiplication, counit and antipode maps of theHopf algebra U by ∆ , (cid:15) and S , respectively. The axiom implies that U is a Λ-graded Hopf algebra in which C [Λ ∗ ] is a commutative Hopfsubalgebra in degree 0. In particular, the unit is 1 U = 1 W ⊗ ∈ W ⊗ C C [Λ ∗ ]. Also, for any homogenous x, y ∈ U , | xy | = | x | + | y | ∈ Λ, | x (1) | + | x (2) | = | x | where ∆ x = (cid:80) x (1) ⊗ x (2) and finally S ( x ) = gxg − .We can now build the unrolled version U H of U : Let S H be thetensor symmetric algebra of H which can be identified with polynomialmaps on H ∗ = Λ ⊗ Z C . It has a Hopf algebra structure with elementsof H being primitive. Consider the semi-direct product(2) U H = U (cid:111) S H where for each a ∈ Λ ∗ the action of the associated element H a ∈ S H on a homogeneous element x ∈ U is given by [ H a , x ] = a ( | x | ) x orequivalently(3) H a x = x ( H a + a ( | x | )) . Proposition 2.1.
The Hopf algebra morphisms of U and S H naturallyextend (via multiplication) to comultiplication ∆ , counit (cid:15) and antipode S maps on U H , making U H into a Λ -graded pivotal Hopf algebra withthe same pivot g as in U .Proof. To see that U H is a Hopf algebra we need to check that theextended maps satisfy the relations given in Equation (3). For example,it is easy to check that∆( H a x ) = ∆( H a )∆( x ) = ∆( x )∆( H a )+∆( x ) a ( | x | ) = ∆( x ( H a + a ( | x | ))) . ODIFIED GRADED HENNINGS INVARIANT 5
Since g is grouplike ∆( g ) = g ⊗ g so its Λ-degree is zero. So theconjugation by g is trivial on S H , as is the square of the antipode. (cid:3) Remark 2.2.
Suppose that Λ (cid:48) is a rank r sub-lattice of Λ ∗ such that U Λ (cid:48) = W ⊗ C C [Λ (cid:48) ] ⊂ U is a sub-Hopf algebra. Then U H Λ (cid:48) = U Λ (cid:48) (cid:111) S H is a sub-Hopf algebra of U H . Furthermore, if the pivot g ∈ U is in U Λ (cid:48) then g is also a pivot for U Λ (cid:48) and U H Λ (cid:48) . We say that U Λ (cid:48) and U H Λ (cid:48) are Λ (cid:48) versions of U and U H .We call U H an unrolled Hopf algebra . Example 2.3. (Continued in Examples 2.7, 4.17 and 4.22). In thisexample we consider the case of quantum sl . Assume (cid:96) is greater than3 and let (cid:96) (cid:48) = (cid:96)/gcd ( (cid:96), U be the C -algebra with generators E , F , K and K − with relations KK − = K − K = 1 , KEK − = qE, KF K − = q − F, [ E, F ] = K − K − q − q − , E (cid:96) (cid:48) = 0 , F (cid:96) (cid:48) = 0 . (4)The algebra U is a Hopf algebra where the coproduct, counit and an-tipode are defined by∆( E ) = 1 ⊗ E + E ⊗ K , ε ( E ) = 0 , S ( E ) = − EK − , ∆( F ) = K − ⊗ F + F ⊗ , ε ( F ) = 0 , S ( F ) = − K F, ∆( K ) = K ⊗ K ε ( K ) = 1 , S ( K ) = K − . The Hopf algebra U is pivotal with pivot g = K − (cid:96) (cid:48) . The sub-Hopfalgebra U ξ ( sl ) generated by E, F and K ± is known as the semi-restricted quantum group where the word semi-restricted is used be-cause E (cid:96) (cid:48) = F (cid:96) (cid:48) = 0.Let U H be the C -algebra given by generators E, F, K ± , H and rela-tions in Equation (4) plus the relations: HK = KH, [ H, E ] = E, [ H, F ] = − F. The algebra U H is a Hopf algebra where the coproduct, counit andantipode are defined by the above equations for E, F and K and∆( H ) = H ⊗ ⊗ H, ε ( H ) = 0 , S ( H ) = − H. The sub-Hopf algebra generated by
E, F, K ± and H is known as U Hξ ( sl ) and called the unrolled quantum group associated to sl .We now explain that our construction recovers these quantum groups.Let Λ be the rank one free abelian group Z . Let W be the Λ-graded C -vector space with basis F i E j for 0 ≤ i, j < (cid:96) (cid:48) and grading given by | F i E j | = j − i . Here C [Λ ∗ ] is identified with Laurent polynomials inthe variable K , which as above is a Hopf algebra where each element In most of the litterature, including [6, 12, 14], the element K and 2 H arecalled K and H respectively. NATHAN GEER, NGOC PHU HA, AND BERTRAND PATUREAU-MIRAND is group like. As a vector space U is isomorphic to W ⊗ C C [Λ ∗ ] andsatisfies Axiom 1. Let Λ (cid:48) = 2Λ ∗ then U ξ ( sl ) = U Λ (cid:48) where U Λ (cid:48) is theΛ (cid:48) version of U , see Remark 2.2. Moreover, S H is C [ H ] and the Hopfalgebra of Proposition 2.1 is isomorphic to U H where U Hξ ( sl ) = U H Λ (cid:48) isits Λ (cid:48) version.More generally, we have: Example 2.4. (Continued in Examples 2.8, 4.18 and 4.22). Here let (cid:96) be odd and greater than 3. Let g be a simple finite dimensionalcomplex Lie algebra of rank r with a set of simple roots { α , ..., α r } . Let A = ( a ij ) ≤ i,j ≤ r be the Cartan matrix corresponding to these simpleroots. Consider the unrolled quantum group U Hξ ( g ) associated to g ,given in [12]. This algebra has generated denoted K β , X i , X − i and H α i where β is in the root lattice Λ and 1 ≤ i ≤ r (for the relations see[12]). The semi-restricted quantum group U ξ ( g ) is the subalgebra of U Hξ ( g ) generated by K β , X ± i for all β and i .To describe this example we will need the following notation here andlater. There exists a diagonal matrix D = diag ( d , ..., d r ) such that DA is symmetric and positive-definite ( D is unique if 1 ∈ { d i } i ⊂ Z ). Let Λbe the root lattice which is the Z -lattice generated by the simple roots { α i } and let { α ∗ i } be the dual basis of Λ ∗ . Let H = Λ ∗ ⊗ Z C be theCartan subalgebra of g and B : H ∗ × H ∗ → C be the symmetric bilinearform defined by B ( α i , α j ) = d i a ij . For λ ∈ Λ let B λ = B ( λ, · ) ∈ C [Λ ∗ ]and Λ (cid:48) = { B λ : λ ∈ Λ } . Define H α i = (cid:80) j a ij H α ∗ j so that the lattice Λ (cid:48) also identify with the free group generated by the elements { d i H α i , i =1 , . . . , r } in H .To describe W , let β , ..., β N ∈ Λ be an ordering of the set of positiveroots where N = dim( g ) − r . For each i = 1 , ..., N , let X ± β i be the positive(resp. negative) root vector of U ξ ( g ) (see for example [5, Section 8.1and 9.1]). Let W be the Λ-graded C -vector space with homogeneousbasis X i β X i β ...X i N β N X j − β X j − β ...X j N − β N with grading (cid:80) rk =1 ( i k − j k ) β k for i , ..., i N , j , ..., j N ∈ { , ..., (cid:96) − } .The vector space U ξ ( g ) (cid:39) W ⊗ C C [Λ (cid:48) ] embeds into U = W ⊗ C C [Λ ∗ ].The Hopf algebra structure of U ξ ( g ) extends uniquely to a Hopf algebrastructure on U such that relations (1) hold where | X ± i | = ± α i . Then U satisfies Axiom 1 with pivot given by g = B − (cid:96) ) ρ where ρ is thehalf sum of all positive roots (see [12]). Finally, U Λ (cid:48) = U ξ ( g ) and U H Λ (cid:48) = U Hξ ( g ) are the Λ (cid:48) version of the quantum group U and U H .2.2. Category of weight modules.
In the following, we will use thenotation ξ x := exp (cid:0) iπx(cid:96) (cid:1) = (cid:80) ∞ n =0 1 n ! (cid:0) iπx(cid:96) (cid:1) n for x a complex number oran element of a topological algebra. ODIFIED GRADED HENNINGS INVARIANT 7
A finite dimensional U H -module V is a weight module if it is a semi-simple module over the subalgebra S H and(5) K a = ξ H a as operators on V , for any a ∈ Λ ∗ . Let C H be the tensor categoryof U H -weight modules. The eigenspaces for the action of S H on aweight module V are called weight spaces of V and this action gives a H ∗ -grading on V .Recall that a pivotal category is a tensor category with left duality { ←− coev V , ←− ev V } V and right duality { −→ coev V , −→ ev V } V which satisfy certaincompatibility conditions, see for example [2]. The category C H is apivotal category with duality maps: ←− coev V : C → V ⊗ V ∗ , given by 1 (cid:55)→ (cid:88) v i ⊗ v ∗ i , ←− ev V : V ∗ ⊗ V → C , given by f ⊗ w (cid:55)→ f ( w ) , −→ coev V : C → V ∗ ⊗ V, given by 1 (cid:55)→ (cid:88) v ∗ i ⊗ g − v i , −→ ev V : V ⊗ V ∗ → C , given by w ⊗ f (cid:55)→ f ( gw )where { v i } is a basis of V and { v ∗ i } is its dual basis of V ∗ = Hom C ( V, C ).Let(6) G = H ∗ / Λ (cid:39) ( C / Z ) r then C H is G -graded: a module of C H is homogeneous of degree ¯ α ∈ G if all its weights belong to ¯ α . We call C H ¯ α the full subcategory of degree¯ α homogeneous modules. One easily check that any module of C H isa direct sum of homogeneous module and that the Hom set of twohomogeneous module of different degrees is zero. We summarize thisby writing(7) C H = (cid:77) ¯ α ∈ G C H ¯ α Given an element Q = (cid:88) i c i ⊗ c (cid:48) i ∈ Λ ∗ ⊗ Z Λ ∗ ⊂ H ⊗ H ⊂ U H ⊗ U H we define the following four maps. First, let B : H ∗ × H ∗ → C bethe symmetric bilinear form given by the element Q , in particular, B ( λ, µ ) = Q ( λ ⊗ µ ) ∈ Z for ( λ, µ ) ∈ Λ . Second, if V, W are H ∗ -gradedvector space then let H V,W = ξ Q : V ⊗ W → V ⊗ W be the operatordefined by H V,W ( v ⊗ w ) = ξ B ( | v | , | w | ) v ⊗ w for homogeneous vectors v and w . Third, consider the map B : Λ → Λ ∗ ⊂ C [Λ ∗ ], λ (cid:55)→ B λ where B λ = B ( λ, · ). Then we have B λ B µ = B λ + µ in C [Λ ∗ ] for λ, µ ∈ Λ. Usingthis map we can define an outer automorphism (cid:101) H of U H ⊗ U H givenby(8) (cid:101) H ( x ⊗ y ) = xB | y | ⊗ B | x | y. NATHAN GEER, NGOC PHU HA, AND BERTRAND PATUREAU-MIRAND for x, y ∈ U H . This outer automorphism is compatible with conjuga-tion by H in C H : if ρ V i : U H → End C H ( V i ), for i = 1 ,
2, are objects in C H then( ρ V ⊗ ρ V )( (cid:101) H ( x ⊗ y )) = H V ,V ( ρ V ( x ) ⊗ ρ V ( y )) H − V ,V for x, y ∈ U H .Throughout this paper, for two spaces X and Y we denote their flipmap as τ : X ⊗ Y → Y ⊗ X which given by x ⊗ y (cid:55)→ y ⊗ x . We call anelement(9) ˇ R = (cid:88) i x i ⊗ y i ∈ U ⊗ U a quasi R-matrix for Q if it satisfies the Relations ( ˇR1)–( ˇR4) givenbelow: τ (∆ u ) = (cid:101) H (cid:0) ˇ R (∆ u ) ˇ R − (cid:1) , for all u ∈ U H , ( ˇR1) ∆ ˇ R = (cid:101) H − ( ˇ R ) ˇ R , ( ˇR2) ∆ ˇ R = (cid:101) H − ( ˇ R ) ˇ R , ( ˇR3)where ˇ R = (cid:80) j U H ⊗ x j ⊗ y j , ˇ R = (cid:80) j x j ⊗ y j ⊗ U H , (cid:101) H − ( ˇ R ) = (cid:80) j x j ⊗ ( B | y j | ) − ⊗ y j and (cid:101) H − ( ˇ R ) = (cid:80) j x j ⊗ ( B | x j | ) − ⊗ y j . Finally,we require ˇ R is compatible with the pivot g by assuming( ˇR4) (cid:88) i y i gx i B | x i | = (cid:88) i x i g − y i ( B | x i | ) − . Note, above we assume the expression of ˇ R = (cid:80) j x j ⊗ y j is given withhomogeneous elements x i , y i ∈ U . Axiom 2.
Assume Axiom 1 and U H has an element ˇ R which is a quasiR-matrix for Q . In the rest of the paper, we assume Axiom 2 is true. Then Hopfalgebra U H is not quasi-triangular as H / ∈ U H ⊗ U H but as we will seenext the category C H is still braided even ribbon.Recall a braiding on a tensor category C consists of a family ofnatural isomorphisms { c V,W : V ⊗ W → W ⊗ V } satisfying the HexagonAxiom: c U,V ⊗ W = (Id V ⊗ c U,W ) ◦ ( c U,V ⊗ Id W ) c U ⊗ V,W = ( c U,W ⊗ Id V ) ◦ (Id U ⊗ c V,W )for all
U, V, W ∈ C . We say C is braided if it has a braiding. If C ispivotal and braided, one can define a family of natural automorphisms θ V = (Id ⊗ −→ ev V )( c V,V ⊗ Id)( v ⊗ ←− coev V ) : V → V. Following [13], we say that C is ribbon and the morphism θ is a twist if(10) θ V ∗ = ( θ V ) ∗ ODIFIED GRADED HENNINGS INVARIANT 9 for all V ∈ C . Proposition 2.5.
Category C H is ribbon with pivotal structure givenabove and braiding given by c V,W = τ ◦ H ◦ ˇ R : V ⊗ W → W ⊗ V .Proof. Axioms ( ˇR1)–( ˇR4) imply that for weight modules
V, V , V , W ,( ˇR1) = ⇒ c V,W : V ⊗ W → W ⊗ V is a morphism in C H ,( ˇR2) = ⇒ c V ⊗ V ,W = ( c V ,W ⊗ Id)(Id ⊗ c V ,W ),( ˇR3) = ⇒ c W,V ⊗ V = (Id ⊗ c W,V )( c W,V ⊗ Id),( ˇR4) = ⇒ the pivotal structure and the braiding are compatible i.e. θ V := (Id ⊗ −→ ev V )( c V,V ⊗ Id)(Id ⊗ ←− coev V ) is a twist in C H .We justify with more details this last sentence. The pivotal structuregives natural isomorphisms V ∗∗ ∼ = V . Using them, the dual and equiv-alent equality to (10) is given by θ (cid:48) V = θ V where θ (cid:48) V = ( ←− ev V ⊗ Id)(Id ⊗ c V,V )( −→ coev V ⊗ Id) . Recall that −→ ev V = ←− ev V ◦ τ ◦ ( g ⊗
1) and −→ coev V = (Id ⊗ g − ) ◦ τ ◦ ←− coev V .As | g | = 0, the pivot commute with H . Similarly, as (cid:101) H and τ are trivialon ∆( H ), then Relation ( ˇR1) implies that ˇ R commute with ∆( H ) andthus | x i | + | y i | = 0. Then we can compute for a vector v ∈ V withweight λ : θ V ( v ) = (Id ⊗ −→ ev V )( c V,V ⊗ Id)( v ⊗ ←− coev V )= (Id ⊗ ←− ev V τ )(Id ⊗ ρ ( g ) ⊗ Id)( τ H V,V ρ ⊗ ( ˇ R ) ⊗ Id)( v ⊗ ←− coev V )= (cid:88) i (Id ⊗ ←− ev V τ )( τ H V,V ⊗ Id) (cid:16) ρ ( gx i )( v ) ⊗ ( ρ ( y i ) ⊗ Id)( ←− coev) (cid:17) = (cid:88) i ξ B ( | x i | + λ,λ ) (Id ⊗ ←− ev V τ )( τ ⊗ Id) (cid:16) ρ ( gx i )( v ) ⊗ ( ρ ( y i ) ⊗ Id)( ←− coev) (cid:17) = (cid:88) i ξ B ( | x i | + λ,λ ) ( ρ ( y i ) ⊗ ←− ev V )(Id ⊗ τ )( τ ⊗ Id) (cid:16) ρ ( gx i )( v ) ⊗ ←− coev (cid:17) = (cid:88) i ξ B ( | x i | + λ,λ ) ( ρ ( y i ) ⊗ ←− ev V ) (cid:16) ←− coev ⊗ ρ ( gx i )( v ) (cid:17) = ξ B ( λ,λ ) (cid:88) i ξ B ( | x i | ,λ ) ρ ( y i gx i )( v ) . Here the third equality is because ρ ( g ) commutes with H V,V , the fourthis because θ V is H -equivariant so θ V ( v ) has weight λ . This implies thatthe operator H V,V is applied on vectors of weight λ on the right (and | gx i v | on the left). Finally the last equality is the zig-zag relation.Similarly, we compute(11) θ (cid:48) V ( v ) = (cid:88) i ξ B ( λ, | y i | + λ ) ρ ( x i g − y i )( v ) = ξ B ( λ,λ ) (cid:88) i ξ − B ( | x i | ,λ ) ρ ( x i g − y i )( v ) and the equality θ V ( v ) = θ (cid:48) V ( v ) follows from Relation ( ˇR4) with B | x i | v = ξ B ( | x i | ,λ ) v . (cid:3) Remark 2.6.
Suppose that Λ (cid:48) is a rank r sub-lattice of Λ ∗ as in Remark2.2. Then one easily check that the analogous category of U H Λ (cid:48) -weightmodule is indentified with C H . Indeed the restriction functor gives any U H -weight module a structure of U H Λ (cid:48) -weight module but reciprocally,Condition (5) gives a unique way to extend any U H Λ (cid:48) -weight module toa U H -module. Notation:
For q ∈ C \ { } and j ∈ N , we use[ j ; q ] = 1 − q j − q and [ j ; q ]! = [ j ; q ] · · · [1; q ] . Example 2.7.
This example builds upon Example 2.3 and is continuedin Examples 4.17 and 4.22. Recall (cid:96) ≥ (cid:96) (cid:48) = (cid:96)/gcd ( (cid:96),
2) and U is adegree 2 extension of the semi-restricted quantum group U Λ (cid:48) = U ξ ( sl ).Then a quasi R-matrix for Q = 2 H ⊗ H is given byˇ R = (cid:96) (cid:48) − (cid:88) j =0 ( ξ − ξ − ) j [ j ; ξ − ]! E j ⊗ F j ∈ U ⊗ U , and if v ⊗ w ∈ V ⊗ W ∈ C H where v, w have weights | v | and | w | , then H V,W ( v ⊗ w ) = ξ | v | . | w | v ⊗ w . With slight abuse of notation we writethe equality of operator on C H as H = ξ H ⊗ H . In [21], it is shown that ˇ R satisfies Relations ( ˇR1)-( ˇR3). For (cid:96) even, [27]has shown an equivalent form of Relation ( ˇR4). For (cid:96) odd the proof ofExample 2.8 can be applied. Hence for any (cid:96) ≥
3, the quantum group U associated to sl of Example 2.3 satisfies Axiom 2. Example 2.8.
Here we use the notation and build upon Example 2.4(continued in Examples 4.18 and 4.22). Recall U is an extension ofthe semi-restricted quantum group U ξ ( g ) associated to a simple finitedimensional complex Lie algebra g . Let Q be the quadratic elementgiven by Q = (cid:80) i,j d i ¯ a ij H α i ⊗ H α j where (¯ a ij ) is the inverse of theCartan matrix. In the basis { H α ∗ i } i =1 ··· n of H dual to { α i } i =1 ··· n , theform Q is given by (cid:80) i,j d i a ij H α ∗ i ⊗ H α ∗ j . So H and ˇ R ∈ U ⊗ U are givenby(12) H = ξ (cid:80) i,j d i a ij H α ∗ i ⊗ H α ∗ j , ˇ R = N (cid:89) i =1 (cid:32) (cid:96) − (cid:88) j =0 (cid:0) ( q β i − q − β i ) X β i ⊗ X − β i (cid:1) j (cid:2) j ; q − β i (cid:3) ! (cid:33) , where q β i = ξ B ( β i ,β i ) / . In [12], ˇ R is shown to satisfy Relations ( ˇR1)-( ˇR3). The Relation ( ˇR4) follows from the fact that Proposition 2.5holds for quantum groups as shown in [13, Theorem 19]: in particular ODIFIED GRADED HENNINGS INVARIANT 11 this implies that θ (cid:48) V = θ V for any module V ∈ C H (see Equation(11)). This equality can be interpreted as follows. Let x (resp. x (cid:48) )be the element on the left (resp. right) side of Equation ( ˇR4). Then θ (cid:48) V = θ V implies that ρ V ( x − x (cid:48) ) = 0 where ρ V : U H → End C H ( V ) isthe representation V . Finally, Proposition 4.21 implies that x − x (cid:48) = 0which gives the desired result. Thus, the quantum group U associatedto g of Example 2.4 satisfies Axiom 2.3. Topological unrolled quantum groups
Unrolled quantum groups are not ribbon Hopf algebras but theyadmit a ribbon category of representations. The goal of this sectionis to define a topology on an unrolled quantum group U H so that itscompletion (cid:99) U H is a topological ribbon Hopf algebra. This topology isthe one of uniform convergence on compact sets induced by the Cartanpart of the algebra. The topology is nice because it produces explicittopological bases for (cid:99) U H and allows the identification of C H with finitedimensional (cid:99) U H weight modules. The topological Hopf algebras areHopf algebra objects in the category of nuclear spaces. We first recallsome definitions from [28, 16, 17].3.1. Topological Hopf algebras.
Let E and F be locally convexspaces. A topology is compatible with ⊗ if both 1) ⊗ : E × F → E ⊗ F is continuous and 2) for all ( e, f ) ∈ E (cid:48) × F (cid:48) the linear form e ⊗ f : E ⊗ F → C , x ⊗ y (cid:55)→ e ( x ) f ( y ) is continuous [28, 16]. The locallyconvex space E is called nuclear , if all the compatible topologies on E ⊗ F agree after completion, for all locally convex spaces F , i.e. thetopology on the completion of E ⊗ F compatible with ⊗ is unique.For two nuclear spaces E and F the completion of the tensor product E ⊗ F endowed with its compatible topology is denoted E (cid:98) ⊗ F . Werecall some facts about nuclear spaces (see [16]): 1) a finite dimensionalspace is nuclear, 2) the tensor product of two nuclear spaces is a nuclearspace and 3) a space is nuclear if and only if its completion is nuclear.Complete nuclear spaces form a symmetric monoidal category Nuc with the product (cid:98) ⊗ (see [28]). Recall we denote the flip isomorphismby τ : E (cid:98) ⊗ F ∼ → F (cid:98) ⊗ E .A topological Hopf algebra is a Hopf algebra object in the monoidalcategory Nuc . That is a complete nuclear C -space A endowed with the C -linear maps called the product, unit, coproduct, counit and antipode m : A (cid:98) ⊗A → A , η : C → A , ∆ : A → A (cid:98) ⊗A , ε : A → C and S : A → A which satisfy the axioms:(1) the product m is associative on A admitting 1 A = η (1) as unity,(2) ( ε (cid:98) ⊗ Id A ) ◦ ∆ = (Id A (cid:98) ⊗ ε ) ◦ ∆ = Id A and the coproduct ∆ iscoassociative: (∆ (cid:98) ⊗ Id A ) ◦ ∆ = (Id A (cid:98) ⊗ ∆) ◦ ∆, (3) ∆ and ε are algebra morphisms where the associative productin A (cid:98) ⊗A is determined by ( m (cid:98) ⊗ m ) ◦ (Id A (cid:98) ⊗ τ (cid:98) ⊗ Id A ),(4) m ◦ ( S (cid:98) ⊗ Id A ) ◦ ∆ = m ◦ (Id A (cid:98) ⊗ S ) ◦ ∆ = η ◦ ε .If V is a finite dimensional C -vector space we denote by C ω ( V )the space of entire functions on V endowed with the topology of uni-form convergence on compact sets. Then C ω ( V ) is a complete nuclearspace. If { Z i } i =1 ,...,n are the coordinate functions of V in some ba-sis then we denote C ω ( V ) by C ω ( Z , . . . , Z n ). Remark that we have C ω ( V ) (cid:98) ⊗ C ω ( V ) (cid:39) C ω ( V × V ) where V , V are finite dimensional C -vector spaces (see [31, Theorem 51.6]).We will need the following example later. Recall the vector spaces W and H given in Section 2. The nuclear space W (cid:98) ⊗ C ω ( H ∗ ) can beseen as W -valued entire functions, whose elements are power series inthe variables ( H i ) i =1 ··· n with coefficients in W . Alternatively, if B W is a base of W , one can think of W (cid:98) ⊗ C ω ( H ∗ ) as the span of B W withcoefficients in C ω ( H ∗ ). Proposition 3.1 ([17]) . Let H , H , W , W be finite dimensional C -vector spaces. Then ( W ⊗ C ω ( H ∗ )) (cid:98) ⊗ ( W ⊗ C ω ( H ∗ )) (cid:39) ( W ⊗ W ) ⊗ C ω ( H ∗ × H ∗ ) . Topology on the completion of U H . The space of entire func-tions is a nuclear space obtained as the completion of polynomial func-tions for the topology of uniform convergence on compact sets. We usea similar completion to define a topological ribbon Hopf algebra from U H .Recall that as a vector space,(13) U H = W ⊗ C C [Λ ∗ ] ⊗ C S H where W is a finite dimensional C -vector space, C [Λ ∗ ] = C [ K ± , . . . , K ± r ]is the space of Laurent polynomials in r variables and S H = C [ H , . . . , H r ]is the space of polynomials in r variables (here we write H i for H a i and K i for K a i , where ( a i ) i is the fixed basis of Λ ∗ ).Denote C ω ( H ∗ ) by C ω ( H , . . . , H r ). We embed the commutativealgebra C [Λ ∗ ] ⊗ C S H into C ω ( H , . . . , H r ) by seeing H i as a linearfunction on H ∗ and by sending K i = K a i to ξ H i = exp( iπ(cid:96) H i ). Then U H is embedded in W ⊗ C C ω ( H ∗ ).Now we put on C ω ( H ∗ ) the topology of uniform convergence on com-pact sets. Then it is a complete nuclear space and as W is finite di-mensional, one has W ⊗ C C ω ( H ∗ ) = W (cid:98) ⊗ C C ω ( H ∗ ) . Furthermore the embedding of U H in this complete nuclear space isdense in it so we have:(14) (cid:99) U H = W (cid:98) ⊗ C ω ( H ∗ ) (cid:39) W ⊗ C C ω ( H , . . . , H r ) . ODIFIED GRADED HENNINGS INVARIANT 13
Let us now describe a family of semi-norms that generates the topol-ogy of (cid:99) U H . Any two norms on W are equivalent. Let choose a norm (cid:107)·(cid:107) on W . Then for any compact set C in H ∗ , we have a semi-norm (cid:107)·(cid:107) C on W ⊗ C C ω ( H ∗ ) defined as follows: If ϕ ∈ C and x is a multivariable powerseries such that x ( H , . . . , H r ) ∈ C ω ( H ∗ ) then ϕ ∗ x ( H , . . . , H r ) is theevaluation of x at ϕ , that is ϕ ∗ x ( H , . . . , H r ) = x ( ϕ ( H ) , . . . , ϕ ( H r )) ∈ C . Then for u = (cid:80) k w k x k ( H , . . . , H r ) ∈ (cid:99) U H the semi-norm of u asso-ciated to C is defined by(15) (cid:107) u (cid:107) C = sup ϕ ∈ C (cid:107) ϕ ∗ u (cid:107) where ϕ ∗ u = (cid:80) k w k ϕ ∗ x k ∈ W . One easily sees that (cid:107)·(cid:107) C does notdepend of the norm on W up to equivalence. The family of semi-norms {(cid:107)·(cid:107) C } C compact generates the topology of uniform convergenceon compact sets of W -valued entire functions.The rest of this subsection is dedicated to proving that the completenuclear space (cid:99) U H has a natural structure of a topological ribbon Hopfalgebra. This means that (cid:99) U H is a topological Hopf algebra with aninvertible element R ∈ U H (cid:98) ⊗U H called the universal R-matrix and acentral element θ ∈ (cid:99) U H called the twist R ∆( x ) = ∆ op ( x ) R for all x ∈ U H , (R1) ∆ ⊗ Id( R ) = R R , (R2) Id ⊗ ∆( R ) = R R , (R3) S ( θ ) = θ, (R4) ∆( θ ) = R R ( θ ⊗ θ ) , (R5) ε ( θ ) = 1 . (R6)Recall the element Q = (cid:80) i c i ⊗ c (cid:48) i ∈ H ⊗ H and define H = ξ Q ∈ U H (cid:98) ⊗U H . Theorem 3.2. (cid:99) U H is a topological ribbon Hopf algebra with universalR-matrix R = H ˇ R ∈ U H (cid:98) ⊗U H and twist (16) θ = g − (cid:0) m ◦ ( S ⊗ Id)( R ) (cid:1) ∈ (cid:99) U H . Remark that as for ribbon Hopf algebras, following the lines of [21, § VIII] we can prove that ( (cid:99) U H , R , θ ) satisfies the following additionalproperties: R R R = R R R , ( ε ⊗ Id)( R ) = 1 = (Id ⊗ ε )( R ) , ( S ⊗ Id)( R ) = R − = (Id ⊗ S − )( R ) , R = ( S ⊗ S )( R ) . In the literature the inverse of the element θ is often considered. To prove the theorem, we need the following three lemmas whichimply that the product, coproduct and antipode are continuous.As W is finite dimensional, all norms on W are equivalent. Nev-ertheless to prove the theorem, we fix a convenient choice of norm asfollows: let B W = ( w , . . . , w n ) be a basis of W where w i is homoge-neous of degree λ i ∈ Λ. Then let (cid:107)·(cid:107) be the maximum norm in B W ,which means (cid:107) (cid:80) i x i w i (cid:107) = sup i | x i | . Similarly, B ⊗ kW is a basis of W ⊗ k that will be equipped with the associated maximum norm. Then oneeasily checks that (cid:107) w ⊗ w (cid:48) (cid:107) = (cid:107) w (cid:107) (cid:107) w (cid:48) (cid:107) .Let E , F be nuclear spaces and let N F be a set of semi-norms on F that generate its topology. Then remark that a linear map f : E → F is continuous if and only if for any continuous semi-norm (cid:107)·(cid:107) F ∈ N F ,there exists a continuous norm (cid:107)·(cid:107) E on E and a constant η ∈ R + suchthat(17) ∀ x ∈ E, (cid:107) f ( x ) (cid:107) F ≤ η (cid:107) x (cid:107) E . Lemma 3.3.
For each compact set C ⊂ H ∗ , there exists a compact set C (cid:48) and a constant λ C ∈ R such that ∀ x, y ∈ U H , we have (cid:107) xy (cid:107) C ≤ λ C (cid:107) x ⊗ y (cid:107) C (cid:48) × C . Proof.
Let Λ W = { λ i , i = 1 · · · N } which is a finite set and C (cid:48) = C +Λ W which is also compact. Then we write ϕ ∗ ( xy ) = ϕ ∗ (cid:32)(cid:88) i w i x i ( H , . . . , H r ) (cid:88) j w j y j ( H , . . . , H r ) (cid:33) = ϕ ∗ (cid:32)(cid:88) i,j w i w j x i ( H + a ( λ j ) , . . . , H r + a r ( λ j )) y j ( H , . . . , H r ) (cid:33) = (cid:88) i,j ϕ ∗ ( w i w j ) . ( ϕ + λ j ) ∗ ( x i ) .ϕ ∗ ( y j ) . Then (cid:107) xy (cid:107) C = sup ϕ ∈ C (cid:107) ϕ ∗ ( xy ) (cid:107) ≤ sup ϕ ∈ C (cid:88) i,j (cid:107) ϕ ∗ ( w i w j ) (cid:107) . | ( ϕ + λ j ) ∗ ( x i ) | . | ϕ ∗ ( y j ) |≤ (cid:88) i,j sup ϕ ∈ C (cid:107) ϕ ∗ ( w i w j ) (cid:107) sup ϕ ∈ C (cid:48) | ϕ ∗ ( x i ) | sup ϕ ∈ C | ϕ ∗ ( y j ) |≤ (cid:107) x (cid:107) C (cid:48) (cid:107) y (cid:107) C (cid:88) i,j (cid:107) w i w j (cid:107) C , and one can take λ C = (cid:80) i,j (cid:107) w i w j (cid:107) C so that (cid:107) xy (cid:107) C ≤ λ C (cid:107) x (cid:107) C (cid:48) (cid:107) y (cid:107) C = λ C (cid:107) x ⊗ y (cid:107) C (cid:48) × C . (cid:3) By Proposition 3.1 we have U H (cid:98) ⊗U H (cid:39) W ⊗ ⊗ C ω ( H i,j ) where H i, = H i ⊗ , H i, = 1 ⊗ H i for i = 1 , . . . , r and the H i,j are seen as coordinates ODIFIED GRADED HENNINGS INVARIANT 15 functions on H ∗ × H ∗ . If C ⊂ H ∗ × H ∗ is compact, we have the associatedseminorm(18) (cid:107) x (cid:107) C = sup ϕ ∈ C (cid:107) ϕ ∗ x (cid:107) where (cid:107)·(cid:107) is the maximum norm in the basis B ⊗ W of W ⊗ . Lemma 3.4.
For each compact set C ⊂ H ∗ × H ∗ , there exists a compactset C ⊂ H ∗ and λ C ∈ R such that ∀ x ∈ U H , we have (cid:107) ∆ x (cid:107) C ≤ λ C (cid:107) x (cid:107) C . Proof.
Let σ : H ∗ × H ∗ → H ∗ be the sum ( ϕ, ϕ (cid:48) ) (cid:55)→ ϕ + ϕ (cid:48) and C = σ ( C )which is a compact in H ∗ . Let x = (cid:80) i w i x i ( H , . . . , H r ) be written inthe basis B W , then (cid:107) ∆ x (cid:107) C = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i (∆ w i ) x i ( H , + H , , . . . , H r, + H r, ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ≤ (cid:88) i sup ϕ ∈ C (cid:107) ϕ ∗ ∆ w i (cid:107) sup ϕ ∈ C | ϕ ∗ x i ( H , + H , , . . . , H r, + H r, ) |≤ (cid:88) i (cid:107) ∆ w i (cid:107) C sup ϕ ∈ C | ϕ ∗ x i ( H , . . . , H r ) | ≤ (cid:107) x (cid:107) C (cid:88) i (cid:107) ∆ w i (cid:107) C , and one can take λ C = (cid:80) i (cid:107) ∆ w i (cid:107) C . (cid:3) Lemma 3.5.
For each compact set C ⊂ H ∗ there exists a compact set C (cid:48) ⊂ H ∗ and a constant λ C such that (cid:107) S ( x ) (cid:107) C ≤ λ C (cid:107) x (cid:107) C (cid:48) for x ∈ U H . Proof.
Let C (cid:48) = − Λ W − C = (cid:83) i − ( λ i + C ) which is compact. Let x = (cid:80) i w i x i ( H , . . . , H r ) be written in the basis B W , then (cid:107) S ( x ) (cid:107) C = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i x i ( − H , . . . , − H r ) S ( w i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:88) i S ( w i ) x i ( − a ( λ i ) − H , . . . , − a r ( λ i ) − H r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C ≤ (cid:107) x (cid:107) C (cid:48) (cid:88) i (cid:107) S ( w i ) (cid:107) C . (cid:3) Proof of Theorem 3.2.
Recall (cid:99) U H is the completion of U H . Therefore,to show that (cid:99) U H is a topological Hopf algebra it is enough to showthe structure morphisms of the Hopf algebra U H are continuous for thetopology of uniform convergence on compact sets: This is obvious forthe unit and follows for the counit as ε ( x ( H , . . . , H r )) = x (0 , . . . , Finally, we need to show that R is a universal R-matrix and θ is atwist. Properties (R1)–(R3) follow from combining the fact that (cid:101) H isthe conjugation by H and ˇ R satisfies Relations ( ˇR1)–( ˇR3).Next we prove Axiom (R4). Let u = (cid:80) i S ( b i ) a i ∈ (cid:99) U H be the Drinfeldelement where R = (cid:80) i a i ⊗ b i ∈ U H (cid:98) ⊗U H . Then following [21, § VIII]we have u is invertible with inverse u − = (cid:80) i b i S ( a i ). Then S ( u − ) = (cid:88) i S ( a i ) S ( b i ) = (cid:88) i S ( a i ) b i = gθ. But R = H ˇ R where ˇ R = (cid:80) i x i ⊗ y i ∈ U ⊗ U . Then (recall x i and y i have opposite degree) R = (cid:88) i (cid:101) H (1 ⊗ y i ) · H · ( x i ⊗
1) = (cid:88) i ( B − | x i | ⊗ y i ) · H · ( x i ⊗ (cid:88) i (1 ⊗ y i ) · H · ( B − | x i | x i ⊗ R = (cid:88) i ( x i ⊗ · H · (1 ⊗ B | x i | y i )thus from (19),(21) S ( u − ) = m ( H ) (cid:88) i B − | x i | S ( x i ) y i = m ( H ) g (cid:88) i x i g − y i B − | x i | and from (20), one gets(22) u − = m ( H ) (cid:88) i B | x i | y i S ( x i ) = m ( H ) g − (cid:88) i y i gx i B | x i | . Hence Relation ( ˇR4) implies that θ = g − S ( u − ) = gu − and property(R4) follows from g − S ( u − ) = S ( u − ) g − = S ( gu − ) = S ( θ ).To prove the last two equalities we follow the proof given in [21, § VIII] for braided Hopf algebras. In particular, the element u satisfies∆( u ) = ( R R ) − ( u ⊗ u ) = ( u ⊗ u )( R R ) − and ε ( u ) = 1 . Then as θ = gu − , we have∆( θ ) = ∆( gu − ) = ∆( g )∆( u − )= ( g ⊗ g )(( R R ) − ( u ⊗ u )) − = ( g ⊗ g )( u − ⊗ u − )( R R )= ( θ ⊗ θ )( R R ) = ( R R )( θ ⊗ θ ) . This proves Equation (R5). The proof of Equation (R6) is similar. (cid:3)
ODIFIED GRADED HENNINGS INVARIANT 17
Category of module.
Let (cid:98) U H -mod be the category of topolog-ical modules (i.e. object of the category of complete nuclear spaceequipped with (cid:98) U H -module maps in this catetegory). Proposition 3.6. C H is a full subcategory of (cid:99) U H -mod.Proof. Since a module V of C H is finite dimensional, its set of weightsis compact and the action of U H : ρ V : U H ⊗ V → V is continuous and extend to a continuous map ρ V : (cid:99) U H ⊗ V → V . (cid:3) Power elements.
RecallΛ (cid:39) Z r , Λ ∗ = Hom Z (Λ , Z ) = ⊕ i Z H i , H ∗ = Λ ⊗ Z C , H = Λ ∗ ⊗ Z C = ⊕ i C H i . Let G = H ∗ / Λ (cid:39) ( C / Z ) r and G (cid:48) = H / Λ ∗ (cid:39) ( C / Z ) r . Also, let H ( n ) = ( H ⊗ ⊗ · · · ⊗ ⊕ (1 ⊗ H ⊗ ⊗ · · · ⊗ ⊕ · · · ⊕ (1 ⊗ · · · ⊗ ⊗ H )which is a subset of ( S H ) ⊗ n ⊂ ( U H ) ⊗ n . The elements of H ( n ) are called linear . For i = 1 · · · r and j = 1 · · · n let H i,j be the element of H ( n ) which is H i in the j th direct sum (or tensor product) of H ( n ) . Theelements { H i,j : i = 1 · · · r, j = 1 · · · n } form a basis of H ( n ) . Recall (cid:99) U H (cid:39) W ⊗ C C ω ( H , . . . , H r ) and it follows that( U H ) (cid:98) ⊗ n (cid:39) C ω ( H i,j ) ⊗ C W ⊗ n . An element of ( U H ) (cid:98) ⊗ n the form ξ H for some H ∈ H ( n ) is called powerlinear . The set L n of power linear elements of ( U H ) (cid:98) ⊗ n form a commu-tative Lie group among invertible elements of ( U H ) (cid:98) ⊗ n . Furthermore,we have the following two facts: 1) the elements of L are group-likeand 2) L n = ( L ) ⊗ n . The complex vector space H ( n ) contains a Z -lattice Λ ∗ ( n ) = (cid:76) i,j Z H i,j . The exponential of elements of Λ ∗ ( n ) form amultiplicative subgroup of L n generated by the elements K i,j = ξ H i,j .We say a finite sum (cid:80) α H α H (cid:48) α in ( U H ) ⊗ n , for some H α , H (cid:48) α ∈ Λ ∗ ( n ) is quadratic . Similarly, elements of the set Q n = (cid:110) ξ q : q in ( U H ) ⊗ n is quadratic (cid:111) are called power quadratic . For example, the element H = ξ Q is apower quadratic element in Q .Let us write h • = { H i,j } for the vector of coordinates functions. If q ( h • ) is a quadratic element then let(23) (cid:101) q ( h • , h (cid:48)• ) = q ( h • + h (cid:48)• ) − q ( h • ) − q ( h (cid:48)• ) be the associated integral symmetric bilinear form such that (cid:101) q ( h • , h • ) =2 q ( h • ). For example, if q ( h • ) = H , H , = H ⊗ H and q ( h (cid:48)• ) = H (cid:48) , H (cid:48) , are in Λ ∗ (2) then (cid:101) q ( h • , h (cid:48)• ) = ( H , + H (cid:48) , )( H , + H (cid:48) , ) − H , H , − H (cid:48) , H (cid:48) , = H , H (cid:48) , + H (cid:48) , H , . Remark that the notation q ( h • , h (cid:48)• ) makes easy the change of variables.As such a function can be interpreted as a function on ( H ( n ) ) ∗ × ( H ( n ) ) ∗ ,if ω ∈ ( H ( n ) ) ∗ , we denote q ( ω, h (cid:48)• ) the function q ( h • , h (cid:48)• ) where thevariables h • has been evaluated with ω i.e., we substitute H i,j with ω ( H i,j ) ∈ C .Since q is an integral linear combination of the elements H i,j , wehave Lemma 3.7.
Let q ∈ ( U H ) ⊗ n be quadratic. We have (1) if ω ∈ ( H ( n ) ) ∗ (cid:39) ( H ∗ ) n then (cid:101) q ( ω, h • ) defines an element of H ( n ) , (2) fur-thermore, if ω ∈ Λ n , then (cid:101) q ( ω, h • ) ∈ (Λ ∗ ) n and (3) q induces a map G n → G (cid:48) n given by (24) ¯ ω (cid:55)→ (cid:101) q (¯ ω, h • ) ∈ ( G (cid:48) ) n . Remark 3.8. If l is a linear element, the space ξ l U ⊗ n ⊂ L n U ⊗ n onlydepends on the class l of l in ( G (cid:48) ) n . Indeed, if l (cid:48) and l are in thesame class in ( G (cid:48) ) n then ξ l (cid:48) − l is a unit in U ⊗ n so ξ l (cid:48) − l U ⊗ n = U ⊗ n . Inparticular, if q is quadratic and ¯ ω ∈ G n then ξ (cid:101) q (¯ ω,h • ) U ⊗ n is the space ξ l U ⊗ n for any representant l of (cid:101) q (¯ ω, h • ) ∈ ( G (cid:48) ) n .The following lemma generalizes the conjugation by H given byEquation (8): Lemma 3.9.
Let L = ξ l ∈ L n and Q = ξ q ∈ Q n where l ∈ H ( n ) is alinear element and q ∈ ( U H ) ⊗ n is quadratic. Let x be an homogeneouselement of ( U H ) (cid:98) ⊗ n of weight λ ∈ Λ n . Then LxL − = λ ∗ ( L ) x, and for K = ξ (cid:101) q ( λ,h • ) ∈ ξ Λ ∗ ( n ) ⊂ U ⊗ n we have (25) QxQ − = λ ∗ ( Q ) xK. Proof.
Recall we write h • = { H i,j } , L = ξ l ( h • ) and Q = ξ q ( h • ) = ξ (cid:101) q ( h • ,h • ) where (cid:101) q ( h • , h (cid:48)• ) is the symmetric bilinear form determinedby q . Then from Equation (3) we have Lx = ξ l ( h • ) x = x ξ l ( h • + λ ) = x ξ l ( h • ) ξ l ( λ ) = x L λ ∗ ( L ) , and Qx = x ξ q ( h • + λ ) = x ξ q ( h • ) ξ q ( λ ) ξ (cid:101) q ( λ,h • ) = x Q λ ∗ ( Q ) ξ (cid:101) q ( λ,h • ) . The result follows because Q commutes with ξ (cid:101) q ( λ,h • ) as they are powerelements. (cid:3) ODIFIED GRADED HENNINGS INVARIANT 19
The following proposition records some properties of power elements,the proof follows directly from the previous lemma and the definitionsof power elements.
Proposition 3.10. (1) We have inclusions of algebras: U ⊗ n ⊂ (cid:10) Q n U ⊗ n (cid:11) ⊂ (cid:10) Q n L n U ⊗ n (cid:11) ⊂ ( U H ) (cid:98) ⊗ n , where (cid:104) X (cid:105) means linear combinations of elements of X .(2) The sets Q n U ⊗ n and Q n L n U ⊗ n are stable by permutation of thefactors, by the maps S i = Id ⊗ S ⊗ Id (antipode at the i th position).(3) For ∆ i = Id ⊗ ∆ ⊗ Id and m i = Id ⊗ m ⊗ Id , we have ∆ i (cid:0) Q n U ⊗ n (cid:1) ⊂ Q n +1 U ⊗ n +1 and ∆ i (cid:0) Q n L n U ⊗ n (cid:1) ⊂ Q n +1 L n +1 U ⊗ n +1 ,m i (cid:0) Q n +1 U ⊗ n +1 (cid:1) ⊂ Q n U ⊗ n and m i (cid:0) Q n +1 L n +1 U ⊗ n +1 (cid:1) ⊂ Q n L n U ⊗ n . (4) If V ∈ C H is an homogeneous weight module then Id ⊗ n ⊗ ρ V (cid:0) Q n +1 L n +1 U ⊗ n +1 (cid:1) ⊂ Q n L n U ⊗ n ⊗ C End C ( V ) . G -coalgebra and Discrete Fourier transforms In this section we continue an algebraic investigation of U . Theresults of this section will not be used until Section 6. In particular,Section 5 can be read independently.4.1. Pivotal Hopf G -coalgebra from U . We now recall the defini-tion of Hopf group-coalgebra from [32, 36]. For coherence we will keepan additive notation for the group G . A Hopf G -coalgebra is a fam-ily H = { H α } α ∈ G of algebras over C endowed with a comultiplication∆ = { ∆ α,β : H α + β → H α ⊗ H β } α,β ∈ G , a counit (cid:15) : H → C , and anantipode S = { S α : H α → H − α } α ∈ G which verify some compatibilityconditions, see [26, 33, 36].Let Z = Span (cid:8) K (cid:96)a : a ∈ Λ ∗ (cid:9) = C [ K ± (cid:96) , . . . , K ± (cid:96)r ], recall K i = K a i where { a , . . . , a r } is the fixed basis of Λ ∗ . Then Z is a Hopf subalge-bra of U which is in the center of U . Let Hom Alg ( Z , C ) be the group ofcharacters on Z where the multiplication is given by kh = ( k ⊗ h ) ◦ ∆and k − = k ◦ S | Z . This group is isomorphic to G = (( C / Z ) r , +):Indeed for each ( α , . . . , α r ) ∈ C r denote its image in ( C / Z ) r by ¯ α =( ¯ α , · · · , ¯ α r ). Consider the isomorphism of groups G → Hom
Alg ( Z , C )given by ¯ α = ( ¯ α , · · · , ¯ α r ) (cid:55)→ ( K (cid:96)i (cid:55)→ ξ (cid:96)α i ) for i = 1 , · · · , r . We will usethis isomorphism to identify G with the characters on Z .We now give a Hopf G -coalgebra structure associated to U . For each¯ α = ( ¯ α , · · · , ¯ α r ) ∈ G , consider the ideal I ¯ α (resp. I H ¯ α ) of U (resp. U H )generated by the elements K (cid:96)i − ξ (cid:96)α i for i = 1 , · · · , r . Equivalently, I ¯ α and I H ¯ α are the ideals generated by ( z − ¯ α ( z )) z ∈Z where ¯ α is acharacter under the identification G → Hom
Alg ( Z , C ). The original definition of Hopf group-coalgebra uses more general non abeliangroup.
Let U ¯ α = U /I ¯ α and U H ¯ α = U H /I H ¯ α with the projection maps π ¯ α : U → U ¯ α and π H ¯ α : U H → U H ¯ α . As discussed in Example 2.3 of [18] there exists morphisms ∆ ¯ α, ¯ β and S ¯ α given by the commutative diagrams: U ∆ (cid:45) U ⊗ UU ¯ α + ¯ β π ¯ α + ¯ β (cid:63) ∆ ¯ α, ¯ β (cid:45) U ¯ α ⊗ U ¯ β π ¯ α ⊗ π ¯ β (cid:63) and U S (cid:45) UU ¯ α π ¯ α (cid:63) S ¯ α (cid:45) U − ¯ α .π − ¯ α (cid:63) making U • = {U ¯ α } ¯ α ∈ G a Hopf G -coalgebra. Similarly, using π H ¯ α thereexists ∆ H ¯ α, ¯ β and S H ¯ α making {U H ¯ α } ¯ α ∈ G a Hopf G -coalgebra. Remark that U and its dual are ordinary Hopf algebras.Let us now recall some definitions from [36, 18] Definition 4.1.
Let H • = { H ¯ α } ¯ α ∈ G be a Hopf G -coalgebra.(1) H • is of finite type if for any ¯ α ∈ G , dim C ( H ¯ α ) < ∞ .(2) A G -grouplike element is a family { x ¯ α ∈ H ¯ α } ¯ α ∈ G such that∆ ¯ α, ¯ β ( x ¯ α + ¯ β ) = x ¯ α ⊗ x ¯ β , for all ¯ α, ¯ β ∈ G .(3) A pivot for H • is a G -grouplike element { g ¯ α } ¯ α ∈ G such that S − ¯ α S ¯ α = g ¯ α · g − α , for all ¯ α ∈ G . If H • has a pivot we sayit is a pivotal Hopf G -coalgebra.(4) A right (resp. left) G -integral for H • is a family of linear forms λ R = { λ R ¯ α ∈ H ∗ ¯ α } ¯ α ∈ G (resp. λ L = { λ L ¯ α ∈ H ∗ ¯ α } ¯ α ∈ G ) such that( λ R ¯ α ⊗ Id H ¯ β )∆ ¯ α, ¯ β ( x ) = λ R ¯ α + ¯ β ( x )1 ¯ β , (resp. (Id H ¯ α ⊗ λ L ¯ β )∆ ¯ α, ¯ β ( x ) = λ L ¯ α + ¯ β ( x )1 ¯ α )for all ¯ α, ¯ β ∈ G and x ∈ H ¯ α + ¯ β .(5) H • is unimodular if H is a unimodular Hopf algebra i.e. thereexists a non zero element Υ ∈ H , called a two side cointegral ,such that x Υ = Υ x = ε ( x )Υ for all x ∈ H . From now we make the following assumption for U : Axiom 3.
The Hopf G -coalgebra U • = {U ¯ α , g ¯ α } ¯ α ∈ G is unimodular. Theorem 4.2 (Virelizier [36]) . If ( H • , Υ) is a finite type unimodularHopf G -coalgebra, then (1) the space of right G -integral is -dimensional, (2) the right G -integral { λ R ¯ α } ¯ α ∈ G such that λ R (Υ) = 1 is unique, (3) there exists a unique G -grouplike element γ • called distinguishedsuch that { λ R ¯ α ( γ ¯ α · ) } ¯ α ∈ G is a left G -integral, (4) λ R ¯ α ( γ ¯ α · ) = λ R − ¯ α ◦ S ¯ α and ( S − ¯ α S ¯ α ) = γ ¯ α · γ − α ODIFIED GRADED HENNINGS INVARIANT 21 (5) λ R ¯ α ( xy ) = λ R ¯ α ( S − ¯ α S ¯ α ( y ) x ) for all x, y ∈ H ¯ α .Proof. These results are adapted versions of [36], Theorem 3.6, Theo-rem 4.2 and Lemma 4.6 when the group ( G, +) is commutative and H • is unimodular so that the element called in [36] the distinguish elementof H ∗ simplifies to the counit which acts trivially on H ¯ α . Remark thatthe unimodularity is not needed for (1) and (3). (cid:3) The Hopf G -coalgebras U • = {U ¯ α } ¯ α ∈ G and U H • = {U H ¯ α } ¯ α ∈ G are piv-otal with the pivot given by the image g ¯ α of the pivotal element g of U into U ¯ α (resp. into U H ¯ α ). Moreover, U • is of finite type, thus it has aright G -integral.4.2. Discrete Fourier transforms.
In this subsection we recall thediscrete Fourier transform from [17] that will be used to send the uni-versal invariant into a Hopf G -coalgebra of finite type. The results allowus to construct an invariant of 3-manifolds in the next subsection.Let V be the space ( h ∗ ) m for some m . Denote Z , . . . , Z n as thecoordinate functions of V = ( h ∗ ) m . Recall C ω ( Z , . . . , Z n ) is the set ofentire functions on V which is a subalgebra of ( U H ) (cid:98) ⊗ m . Given a map f : C n → C , we define t i ( f ) on C n by t i ( f )( Z , . . . , Z n ) = f ( Z , . . . , Z i + 1 , . . . , Z n ) for 1 ≤ i ≤ n. Let Λ −→ α = { ( α , . . . , α n ) + Z n } be the lattice of C n corresponding to −→ α = ( ¯ α , . . . , ¯ α n ) ∈ ( C / Z ) n .A function f ( Z , . . . , Z n ) ∈ C ω ( Z , . . . , Z n ) is called (cid:96) -periodic in Z i on the lattice Λ −→ α if it satisfies f | Λ −→ α = t (cid:96)i ( f ) | Λ −→ α . The functions { ξ mZ i } i =1 ,...,nm ∈ Z are (cid:96) -periodic and ξ (cid:96)Z i − ξ (cid:96)α i is zero on Λ −→ α . Let I −→ α bethe ideal generated by ξ (cid:96)Z i − ξ (cid:96)α i for 1 ≤ i ≤ n . For m ≤ n , let C ω [ m ] bethe subring of entire functions that are Laurent polynomial in the first m variables: C ω [ m ] = C ω ( Z m +1 , . . . , Z n )[ ξ ± Z , . . . , ξ ± Z m ] ⊂ C ω ( Z , . . . , Z n ) . Then an element of C ω [ m ] /I −→ α defines a (cid:96) -periodic map in all first m variables on Λ −→ α . Proposition 4.3.
Let f = f ( Z , . . . , Z n ) ∈ C ω ( Z , . . . , Z n ) be a (cid:96) -periodic function on Λ −→ α in the first m variable. Then there is anelement F −→ α ( f ) ∈ C ω [ m ] called the discrete Fourier transform of f whichis unique modulo I −→ α , coincides with f on Λ ¯ α ,..., ¯ α m × C n − m and is givenby (26) F −→ α ,m ( f ) = (cid:96) − (cid:88) k ,...,k m =0 a k ··· k m ξ k Z + ··· + k m Z m . The coefficients a k ··· k m ∈ C ω ( Z m +1 , . . . , Z n ) (Fourier coefficients) aredetermined by a k ··· k m = 1 (cid:96) m (cid:96) − (cid:88) i , . . . , i m = 0 β j = α j + i j ξ − k β −···− k m β m f ( β , . . . , β m , Z m +1 , . . . , Z n ) . Proof.
As proven, in [17, Proposition 4.6], for any fixed ( Z m +1 , . . . , Z n ) ∈ C n − m , the element in Equation (26) is the unique linear combinationof the elements { ξ k Z + ··· + k m Z m : 0 ≤ k i < (cid:96) } which coincide with f for( Z , . . . , Z m ) ∈ Λ ¯ α ,..., ¯ α m . (cid:3) The complex analytic Nullstellensatz Theorem from the work of H.Cartan [4] implies that I −→ α is precisely the ideal of entire functions thatvanish on Λ −→ α . Here we give an elementary proof that was shown to usby David E. Speyer: Proposition 4.4.
Let f ∈ C ω ( Z , . . . , Z n ) be zero on Λ ¯ α ,..., ¯ α m × C n − m then f ∈ I −→ α . To prove this proposition, we need the following lemma:
Lemma 4.5.
Let { ϕ k ∈ C ω ( Z , . . . , Z n ) } k ∈ Z be a family of entire func-tions. Then there exists ϕ ∈ C ω ( Z , . . . , Z n ) such that ϕ k = ϕ ( k, Z , . . . , Z n ) for all k ∈ Z .Proof. Let ϕ ( Z , . . . , Z n ) be of the form ϕ ( y ) + ∞ (cid:88) k =1 (cid:81) k − m =1 ( Z − m ) (cid:81) k − m =1 ( k − m ) (cid:18) a − k k − Z k + a k k + Z k (cid:19) (cid:18) Z k (cid:19) C k for some entire functions a ± k ∈ C ω ( Z , . . . , Z n ) and some positive in-tegers C k which we compute inductively as we will now describe. Notethat, for k > p , the product (cid:81) k − m =1 vanishes at Z = ± p , so termsbeyond the p -th term don’t affect the value of the sum at Z = ± p .Also, at Z = ± k , the k th term of the series reduces to a ± k . So, if wehave chosen a m , b m and C m for m < k , there is a unique choice of a ± k which will make the sum correct at Z = ± k . Make that choice for a ± k .Then choose C k large enough that the k th summand has module lessthan 2 − k on the compact {| Z i | ≤ k − , i = 1 · · · n } for all i = 1 · · · n .(Since Z /k ≤ (1 − /k ) on this compact, taking C k large enoughwill work). Thus, the sum will be uniformly convergent on compactsubsets, and will thus give an entire function. (cid:3) Proof of Proposition 4.4.
First remark that the entire function (cid:0) ξ (cid:96)Z − ξ (cid:96)α (cid:1) ∈ C ω ( Z ) ODIFIED GRADED HENNINGS INVARIANT 23 only has simple zeros at integers. We prove the Proposition by induc-tion on n : for any k ∈ Z , f ( α + k, Z , . . . , Z n ) is zero on the latticeΛ α ,...,α n . Then assume this function belongs to the ideal generated by (cid:0) ξ (cid:96)Z i − ξ (cid:96)α i (cid:1) i =2 ··· n . Then we can write f ( α + k, Z , . . . , Z n ) = n (cid:88) i =2 (cid:0) ξ (cid:96)Z i − ξ (cid:96)α i (cid:1) ϕ i,k ( Z , . . . , Z n )then by the above Lemma applied to the family ( ϕ i,k ) k ∈ Z , we obtainentire functions ϕ i ∈ C ω ( Z , . . . , Z n ) such that for ϕ = n (cid:88) i =2 (cid:0) ξ (cid:96)Z i − ξ (cid:96)α i (cid:1) ϕ i ,f − ϕ vanishes on ( α + Z ) × C n − . But then ψ = f − ϕξ (cid:96)Z − ξ (cid:96)α is anentire function and f = ϕ + (cid:0) ξ (cid:96)Z − ξ (cid:96)α (cid:1) ψ ∈ I −→ α . (cid:3) The following proposition says, modulo the ideal I −→ α , the (cid:96) -periodicelements are equal to their Fourier transform and power quadraticelements reduce to power linear. Here we use the notation z • =( Z , . . . , Z n ) and say a linear element is any complex linear combi-nation of the Z i while a quadratic element is a homogeneous degree 2polynomial in the variables z • with integer coefficients. Proposition 4.6. If f ∈ C ω ( Z , . . . , Z n ) is periodic in its first m variables then f = F −→ α ,m ( f ) modulo I −→ α . Consequently, if q ( z • ) is aquadratic element and l is the linear element l ( z • ) = (cid:101) q ( ˜ α, z • ) for any (cid:101) α = ( α , . . . , α n ) ∈ Λ −→ α , then ξ q ( z • ) ∈ ξ l ( z • ) C [ ξ Z , . . . , ξ Z n ] modulo I −→ α .Proof. As f − F −→ α ,m ( f ) is zero on Λ −→ α then Proposition 4.4 implies itbelongs to I −→ α . To prove the second statement of the proposition, let e i be the vector (0 , . . . , , . . . ,
0) then t (cid:96)i ( l ( z • )) − l ( z • ) = (cid:101) q ( (cid:101) α, (cid:96)e i ) = (cid:96) (cid:101) q ( e i , (cid:101) α ) while t (cid:96)i ( q ( z • )) − q ( z • ) = (cid:101) q ( (cid:96)e i , z • ) + q ( (cid:96)e i ) which on Λ −→ α takesvalues in (cid:96) (cid:101) q ( e i , (cid:101) α ) + (cid:96) Z . So ξ q ( z • ) − l ( z • ) is (cid:96) -periodic in all its variableon Λ −→ α so, modulo I −→ α , it is equal to F −→ α ,n ( ξ q ( z • ) − l ( z • ) ) and ξ q ( z • ) = ξ l ( z • ) F −→ α ,n ( ξ q ( z • ) − l ( z • ) ). (cid:3) Next we would like to apply the above results to the context of thetopological unrolled quantum groups. For −→ α = ( ¯ α , . . . , ¯ α n ) ∈ G n we denote by I H −→ α the ideal of (cid:0) U H (cid:1) (cid:98) ⊗ n generated by the rn elements ξ (cid:96)H i,j − ξ (cid:96) ¯ α i,j where ¯ α i,j = ¯ α j ( H i ) ∈ C / Z is the value of H i,j at −→ α .Recall in Subsection 3.4 we define the power linear L n and quadraticelements Q n . In particular, recall L n = { ξ H : H ∈ H ( n ) } where theelements { H i,j } i =1 ,...,r, j =1 ,...,n are a basis of H ( n ) . Let LU ¯ α be the imageof L U in the quotient U H ¯ α . Proposition 4.7.
Let −→ α = ( ¯ α , . . . , ¯ α m , . . . , ¯ α m + n ) ∈ G m + n and let h • denotes the set of r ( m + n ) variables { H i,j } i =1 ,...,r, j =1 ,...,m + n . Let u ∈ Q m + n L m + n U ⊗ m + n ⊂ C ω ( h • ) ⊗ W ⊗ m + n (cid:39) ( U H ) (cid:98) ⊗ m + n be an (cid:96) -periodic element in the first rm variables { H i,j } i =1 ,...,r, j =1 ,...,m on the lattice Λ −→ α . Then π ( u ) ∈ U ¯ α ⊗ · · · ⊗ U ¯ α m ⊗ LU ¯ α m +1 ⊗ · · · ⊗ LU ¯ α m + n where π : ( U H ) (cid:98) ⊗ m + n → ( U H ) (cid:98) ⊗ m + n /I H −→ α is the projection.Proof. Let us write u = (cid:80) k u k ( h • ) w k in a basis ( w k ) k of W ⊗ m + n . Bythe previous proposition, each u k is equal modulo I H −→ α to a Laurent poly-nomial in C [ ξ ± H i,j ] times a power linear element in the last rn variables ξ (cid:80) β i,j H i,j where the sum range for i = 1 · · · r, j = m + 1 · · · m + n . Thenfor γ j ( H , . . . , H r ) = (cid:80) ri =1 β i,j H i , we have: π ( u ) ∈ U ¯ α ⊗ · · · ⊗ U ¯ α m ⊗ ξ γ m +1 U ¯ α m +1 ⊗ · · · ⊗ ξ γ m + n U ¯ α m + n . (cid:3) The symmetrized G -integral. Recall that we denote by h =( H , . . . , H r ) and h j = ( H ,j , . . . , H r,j ) so for example, H = ξ Q ( h ,h ) .Fix α ∈ H and let ¯ α ∈ G be its class modulo Λ. Then we have (cid:101) Q (( − α, α ) , ( h , h )) = Q ( h − α, h + α ) − Q ( h , h ) + Q ( α, α )= Q ( α, h ) − Q ( α, h ) . So ξ (cid:101) Q (( − α,α ) , ( h ,h )) = L ( h − h ) where L ( h ) = ξ Q ( α,h ) and Propo-sition 4.6 implies that L ( h − h ) H is in C [ ξ h , ξ h ] modulo I ( − α,α ) .In particular, L ( h − h ) H is (cid:96) -periodic in all its variable on Λ ( − ¯ α, ¯ α ) .Let H α ∈ U − ¯ α ⊗ U ¯ α be its Fourier transform on this lattice. Hence H = L α H α modulo I ( − α,α ) where L α = L ( h − h ). Proposition 4.8.
The finite dimensional Hopf algebra U is ribbonwith universal R -matrix R = H ˇ R ∈ U ⊗ U and twist θ = g − ( m ◦ ( S ⊗ Id)( R )) = m ( H ) (cid:80) i y i gx i B | x i | ∈ U .Proof. For α = 0, one has L α = 1 and the image of the R -matrix in U H (cid:98) ⊗U H belongs in fact to U ⊗ U . So R = H ˇ R modulo the ideal I H(cid:126) and R inherits all properties of the R -matrix R . Finally the formulafor the twist comes from θ = gu − and Equation (22). (cid:3) Both the quotient morphism U H → U H and the inclusion morphism U → U H are morphisms of ribbon Hopf algebra.To build a invariant of 3-manifold, we will need the following axiomthat will fix the normalization of the cointegral Υ and integral λ R : Axiom 4.
Let δ = λ R ( θ ) and δ = λ R ( θ − ) . We assume: δδ = 1 . The true assumption in this axiom is the “non-degeneracy” condition δδ (cid:54) = 0. Indeed, we have ODIFIED GRADED HENNINGS INVARIANT 25
Proposition 4.9. If δδ (cid:54) = 0 then there exists a choice of Υ so thatAxiom 4 is satisfied.Proof. If δδ (cid:54) = 0, it has a square root ρ ∈ C such that replacing Υ with ρ Υ will lead to a new right integral equal to ρ λ R for which Axiom 4 issatisfied. (cid:3) For ¯ α ∈ G and ν ∈ Λ, let p ¯ αν : U ¯ α → U ¯ α be the projection on thesubspace of weight ν of U ¯ α . Lemma 4.10.
The cointegral Υ ∈ U and the right G -integral { λ R ¯ α } ¯ α ∈ G are of degree , meaning they satisfy: Υ = p (Υ) and (27) λ R ¯ α = λ R ¯ α ◦ p ¯ α for all ¯ α ∈ G .Proof. For any ν ∈ Λ, let Υ ν = p ν (Υ). Now for any homogeneous x ∈ U of weight | x | , x Υ ν = p | x | + ν ( x Υ) = ε ( x ) p | x | + ν (Υ) = ε ( x )Υ ν , where the last equality is because ε ( x ) = 0 unless | x | = 0. HenceΥ ν is a left cointegral in U . Since the space on left cointegral is onedimensional (see [29, Theorem 10.2.2]), we get that Υ ν = Υ or Υ ν = 0.Finally, as Υ = (cid:80) ν Υ ν , there exists a unique ν ∈ Λ such that Υ = Υ ν .Similarly, for any ¯ α ∈ G and for any ν ∈ Λ, let λ ν ¯ α = λ R ¯ α ◦ p ¯ αν . Then let x ∈ U ¯ α + ¯ β of weight | x | . Then we have λ ν ¯ α ( x (1) ) x (2) = λ R ¯ α (p ¯ αν ( x (1) )) x (2) = λ R ¯ α ( x (1) ) p ¯ β | x |− ν ( x (2) ) = λ R ¯ α + ¯ β ( x ) p ¯ β | x |− ν (1 U ¯ β ) = λ ν ¯ α + ¯ β ( x )1 U ¯ β . Thus λ ν is aright G -integral. Once again, since the space of right G -integral is onedimensional, one has λ ν = λ R or λ ν = 0 and since λ R (Υ) = 1, we have λ R = λ ν .The fact that ν = 0 is now the consequence of Axiom 4 since θ hasweight 0 and λ R ( θ ) (cid:54) = 0. (cid:3) The next property of U • is called unibalanced in [3, 18]: Proposition 4.11.
The distinguished element of U • is the square ofits pivot: γ ¯ α = g α for all ¯ α ∈ G. To prove the proposition, we use the following Lemma (see also [36]):
Lemma 4.12.
Let Υ − ¯ α (1) ⊗ Υ ¯ α (2) = ∆ − ¯ α, ¯ α (Υ) ∈ U − ¯ α ⊗ U ¯ α . Then λ R − ¯ α (Υ − ¯ α (1) )Υ ¯ α (2) = 1 and λ R − ¯ α (Υ − ¯ α (2) )Υ ¯ α (1) = γ ¯ α . Furthermore for any x ∈ U − ¯ α , one has x Υ − ¯ α (1) ⊗ Υ ¯ α (2) = Υ − ¯ α (1) ⊗ S − − ¯ α ( x )Υ ¯ α (2) and Υ − ¯ α (1) x ⊗ Υ ¯ α (2) = Υ − ¯ α (1) ⊗ Υ ¯ α (2) S − ¯ α ( x ) . Proof.
First ( λ R ⊗ Id)(∆ − ¯ α, ¯ α (Υ)) = λ R (Υ)1 = 1 and since λ L = λ R ( γ · )is a left G -integral, (Id ⊗ λ R )(∆ − ¯ α, ¯ α (Υ)) = ( γ ⊗ λ L )(∆ − ¯ α, ¯ α ( γ − Υ)) = γ. Υ − ¯ α (1) x ⊗ Υ ¯ α (2) = Υ − ¯ α (1) x (1) ⊗ Υ ¯ α (2) x (2) S − ¯ α ( x (3) ) =∆ − ¯ α, ¯ α (Υ x (1) )1 ⊗ S − ¯ α ( x (2) ) = Υ − ¯ α (1) ⊗ Υ ¯ α (2) S − ¯ α ( x ) . Similarly, x Υ − ¯ α (1) ⊗ Υ ¯ α (2) = x (1) Υ − ¯ α (1) ⊗ S − − ¯ α ( x (3) ) x (2) Υ ¯ α (2) =1 ⊗ S − − ¯ α ( x (2) )∆ − ¯ α, ¯ α ( x (1) Υ) = Υ − ¯ α (1) ⊗ S − − ¯ α ( x )Υ ¯ α (2) . (cid:3) Proof of Proposition 4.11.
We have γ ¯ α = λ R − ¯ α (Υ − ¯ α (2) )Υ ¯ α (1) = ( λ R − ¯ α ⊗ Id)( R ∆ − ¯ α, ¯ α (Υ) R − )and R ∆ − ¯ α, ¯ α (Υ) R − = L α H α ˇ R ∆ − ¯ α, ¯ α (Υ)( L α H α ˇ R ) − . Now we write the linear element L α given at the beginning of thissubsection as L α = L ( h ) ⊗ L ( − h ) = L ( h ) L ( h ) − where L ( h ) = ξ Q ( α,h ) ∈ L . Consider the application f + : U H − ¯ α ⊗ U H ¯ α → U H ¯ α givenby f + ( x ⊗ y ) = S ( x ) y and f − : U H − ¯ α ⊗ U H ¯ α → U H ¯ α given by f − ( x ⊗ y ) = yS − ( x ). Remark that f − ( R ) = f − (( S ⊗ S )( R )) = u = gθ − and f + ( R − ) = f + (( S ⊗ Id)( R )) = S ( u − ) = gθ .Then from Lemma above, one has: R ∆ − ¯ α, ¯ α (Υ) R − = L α (1 ⊗ f − (cid:0) H α ˇ R (cid:1) )∆ − ¯ α, ¯ α (Υ)(1 ⊗ f + ( ˇ R − H − α )) L − α = L α (1 ⊗ f − (cid:0) L − α R (cid:1) )∆ − ¯ α, ¯ α (Υ)(1 ⊗ f + ( R − L α )) L − α = L α L ( h )(1 ⊗ f − ( R )) L ( h )∆ − ¯ α, ¯ α (Υ) L ( − h )(1 ⊗ f + ( R − )) L ( − h ) L − α = ( L ( h ) ⊗ gθ − L ( h ))∆ − ¯ α, ¯ α (Υ)( L ( − h ) ⊗ L ( − h ) gθ ) . Finally, since λ R is homogeneous of weight 0, only terms with Υ (1) commuting with L ( ± h ) will contribute so: λ R − ¯ α ⊗ Id( R ∆ − ¯ α, ¯ α (Υ) R − ) = λ R − ¯ α ⊗ Id((1 ⊗ gθ − L ( h ))∆ − ¯ α, ¯ α (Υ)(1 ⊗ L ( − h ) gθ ))= f − ( R ) L ( h ) L ( − h ) f + ( R − ) = uS ( u − ) = g . (cid:3) The following terminology first appeared in [3] in the non-gradedcase, then in [18] for the example of special linear superalgebra sl (2 | Definition 4.13.
The symmetrised G -integral of H • is the family oflinear forms µ = { µ ¯ α ∈ U ∗ ¯ α } ¯ α ∈ G defined by µ ¯ α ( x ) = λ ¯ α ( g ¯ α x ) for x ∈ U ¯ α . The symmetrized G -integral µ has the properties of a G -trace in [36].More precisely, we have the properties below: ODIFIED GRADED HENNINGS INVARIANT 27
Proposition 4.14.
These equalities hold (cid:0) µ α ⊗ g β (cid:1) ∆ α,β ( x ) = µ α + β ( x )1 β for x ∈ U α + β , (28) (cid:0) g − α ⊗ µ β (cid:1) ∆ α,β ( x ) = µ α + β ( x )1 β for x ∈ U α + β , (29) µ ¯ α ( xy ) = µ ¯ α ( yx ) for x, y ∈ U ¯ α , (30) µ − ¯ α (cid:0) S ± ± ¯ α ( x ) (cid:1) = µ ¯ α ( x ) for x ∈ U ¯ α . (31) µ ¯ α ( x ) = µ ¯ α (p ¯ α ( x )) for x ∈ U ¯ α . (32) Proof.
These identities are directly obtained from Theorem 4.2 andLemma 4.10 by the change of variable x (cid:55)→ g ¯ α + ¯ β x and by using theproperties of the pivot. (cid:3) The following proposition implies Examples 2.3 and 2.4 satisfy Ax-iom 3:
Proposition 4.15.
Suppose that Λ (cid:48) is a rank r sub-lattice of Λ ∗ as inRemark 2.2 and let U (cid:48) = U Λ (cid:48) / ( I ∩ U Λ (cid:48) ) . If U (cid:48) is unimodular with aweight cointegral, then so is U .Proof. Since Λ (cid:48) has rank r , the quotient Λ / Λ (cid:48) is finite with d elements.Let ( k , . . . , k d ) ∈ Λ d be representatives of these elements, then U (cid:48) ⊂ U and as C -vector space, U = (cid:76) i k i U (cid:48) . Let k Υ = (cid:80) i k i . If Υ (cid:48) is thecointegral of U (cid:48) then one easily check that Υ = k Υ Υ (cid:48) = Υ (cid:48) k Υ is a twoside cointegral for U . In particular, if k ∈ Λ, then k will permuteelements of Λ / Λ (cid:48) and there exists K (cid:48) i ∈ Λ (cid:48) such that kk Υ = (cid:80) i k i K (cid:48) i thus Υ k = k Υ = (cid:80) i k i ε ( K (cid:48) i )Υ (cid:48) = ε ( k )Υ. (cid:3) Since the Λ (cid:48) version of the quantum groups considered in Examples2.3 and 2.4 are unimodular (see [9]), we have:
Corollary 4.16.
The quantum groups U associated to Examples 2.3and 2.4 are unimodular. Thus, the quantum groups U of Examples 2.3and 2.4 satisfy Axiom 3. Example 4.17.
Here we build upon Examples 2.3 (which is contin-ued in Examples 2.7 and 4.22). As shown in [18, section 5.1], thesymmetrized integral does not depend on ¯ α in the PBW basis: Thereexists η ∈ C such that it is given for any 0 ≤ i, j ≤ (cid:96) (cid:48) − ≤ k ≤ (cid:96) − µ ¯ α ( E i F j K k ) = ηδ i,(cid:96) (cid:48) − δ j,(cid:96) (cid:48) − δ k, . Furthermore, Equation (30) and the commutation relations for K implythat the same formula holds for the product of E i , F j and K k in anyorder. Using the Fourier transform, it can be rewritten for any (cid:96) -periodic element x = (cid:80) i,j E i F j ϕ ij ( H ) as µ ¯ α ( x ) = η(cid:96) (cid:96) − (cid:88) k =0 ϕ (cid:96) (cid:48) − ,(cid:96) (cid:48) − ( α + k ) . We will apply this to compute λ ( θ ± ): In the formula of ˇ R , only thelast term will contribute and this term is c E (cid:96) (cid:48) − ⊗ F (cid:96) (cid:48) − where c = ( ξ − ξ − ) (cid:96) (cid:48) − [ (cid:96) (cid:48) − ξ − ]! = − ( − (cid:96) ξ(cid:96) (cid:48) ( ξ − ξ − ) (cid:96) (cid:48) − . Thus λ ( θ ) = µ ( g − S ( u ) − ) = µ ( cK − ξ H E (cid:96) (cid:48) − K (cid:96) (cid:48) − F (cid:96) (cid:48) − K − (cid:96) (cid:48) )= cξ − η(cid:96) (cid:96) − (cid:88) k =0 ξ k − k . This last Gauss sum is 0 if and only if (cid:96) ∈ N (see for example [6]).Similarly, using that u = (cid:80) i S ( b i ) S ( a i ) = m ( H − ) (cid:80) i S ( y i ) S ( x i ),one has λ ( θ − ) = µ ( g − u ) = µ (cid:16) cg − m ( H − ) S ( F (cid:96) (cid:48) − ) S ( E (cid:96) (cid:48) − ) (cid:17) = µ (cid:16) cg − ξ − H ξ − (cid:96) (cid:48) F (cid:96) (cid:48) − ξ − E (cid:96) (cid:48) − K − (cid:96) (cid:48) (cid:17) = c η(cid:96) (cid:96) − (cid:88) k =0 ξ k − k . Thus if (cid:96) / ∈ N , one can choose η = ξ(cid:96)c ( (cid:80) (cid:96) − k =0 ξ k − k ) − and Axiom 4holds with λ ( θ − ) = ξ = λ ( θ ) − . Example 4.18.
Here we build upon Example 2.4 (which is continuedin Examples 2.8 and 4.22) and sketch a proof that Axiom 4 hold forthese examples. To simplify notation in this example we set U (cid:48) = U Λ (cid:48) .First, the restriction of right integral of U to U (cid:48) = U (cid:48) / ( I ∩U (cid:48) ) is a rightintegral for U (cid:48) . Second, R and θ belongs to the subalgebra U (cid:48) . Hence δ can be computed in the small quantum group U (cid:48) . Finally, in [24, 25]it is shown that U (cid:48) is ribbon and factorizable thus [10, Corollary 3.9]implies it is non degenerate and δ (cid:54) = 0.4.4. Modified trace on
Proj ( C H ) . Here we discuss modified traces,or m-trace for short, defined in [15, 11]. If C is a linear pivotal category,let Proj ( C ) be the ideal of projective objects of C . The right partialtrace of an endomorphism f ∈ End C ( V ⊗ V (cid:48) ) is the endomorphismptr( f ) ∈ End C ( V ) given byptr( f ) := (Id V ⊗ −→ ev V (cid:48) ) ◦ ( f ⊗ Id V (cid:48)∗ ) ◦ (Id V ⊗ ←− coev V (cid:48) ) . Following [15, 11], a right m-trace t on Proj ( C ) is a family of linearmaps { t V : End C ( V ) → k | V ∈ Proj ( C ) } satisfying:(1) Cyclicity : t V ( f (cid:48) ◦ f ) = t V (cid:48) ( f ◦ f (cid:48) ) for all objects V, V (cid:48) ∈ Proj ( C )and for all morphisms f ∈ Hom C ( V, V (cid:48) ) and f (cid:48) ∈ Hom C ( V (cid:48) , V );(2) Partial trace : t V ⊗ V (cid:48) ( f ) = t V (ptr( f )) for all objects V ∈ Proj ( C )and V (cid:48) ∈ C and for every morphism f ∈ End C ( V ⊗ V (cid:48) ). ODIFIED GRADED HENNINGS INVARIANT 29
Similarly, using the left partial trace one can define a left m-trace.Now let C be the C -linear category (cid:76) ¯ α ∈ G C ¯ α in which C ¯ α is the cat-egory U ¯ α -mod of finite dimensional left U ¯ α -modules for ¯ α ∈ G . Since U is unimodular then there exist a right m-trace t on Proj ( C ). Moreover,since U • is unibalanced (i.e. Proposition 4.11), then [18, Theorem 1.1]implies the right m-trace t is also a left m-trace and it is determinedby(33) t U ¯ α ( R x ) = µ ¯ α ( x )where R x is the right multiplication by x ∈ U ¯ α seen as a morphism ofleft U ¯ α -module.Recall that C H = (cid:76) ¯ α ∈ G C H ¯ α is the category of weight modules over U H , see Section 2. Since U ⊂ U H , we can consider the forgetful functorobtain by restriction ResF : C H ¯ α → C ¯ α , V H (cid:55)→ V = ResF( V H ) and f ∈ Hom C H ( V H , W H ) (cid:55)→ ResF( f ) ∈ Hom C ( V, W ). Here V H and V have the same underlying vector space, and V is obtained from V H byforgetting the action of H .Call Proj ( C H ) the ideal of projective modules of C H . We now needto assume Axiom 5.
There exists a projective module P of C H with ResF( P ) projective in C . Axiom 5 implies that
Proposition 4.19.
For any projective module P of C H , ResF( P ) isprojective module module of C .Proof. Since P is projective it belongs to the ideal generated by P .But then ResF( P ) belongs to the ideal generated by ResF( P ) whichis the subcategory of projective objects of C . (cid:3) For V H ∈ Proj ( C H ), we define a linear map t V H : End C H ( V H ) → C , f (cid:55)→ t V H ( f ) := t ResF( V H ) (ResF( f )) . Since the forgetful functor is pivotal and it commutes with the rightand left partial traces and we have
Proposition 4.20.
The family { t V H } V H ∈ Proj ( C H ) determines a rightand left m-trace on Proj ( C H ) . We finish by a proposition used to complete Example 2.8 and below:
Proposition 4.21.
Suppose there exists a dense open subset O of G such that the categories C ¯ α and C H ¯ α are semi-simple for any ¯ α ∈ O (i.e. C and C H are generically semi-simple). Also, suppose that for ¯ α ∈ O , every simple module in C ¯ α is the image of a simple module in C H ¯ α . Then (1) If x ∈ U H satifies ρ V ( x ) = 0 for all V ∈ C H then x = 0 . (2) Axiom 5 holds.Proof.
Consider a non zero x ( h ) = (cid:80) x i f i ( h ) ∈ (cid:99) U H seen as a W -valued entire function. Since x is not zero, there exists α ∈ H ∗ suchthat x ( α ) (cid:54) = 0 and C α is semi-simple. Then F α ( x ) (cid:54) = 0 ∈ U ¯ α (cid:39) (cid:76) i End C ( V i ). Lifting one of the V i to C H , one can found a simple mod-ule V ∈ C H such that ρ V ( F α ( x )) (cid:54) = 0. Let v ∈ V be a homogeneousvector of weight α (cid:48) such that F α ( x ) v (cid:54) = 0. F α ( x ) and x have the samevalues when h belongs to { α + (cid:80) i k i α i , ≤ k i ≤ (cid:96) − } thus there exists α (cid:48)(cid:48) ∈ H ∗ such that α (cid:48)(cid:48) − α (cid:48) = (cid:96)λ ∈ (cid:96) Λ and F α ( x )( α (cid:48)(cid:48) ) = x ( α (cid:48)(cid:48) ). Now let σ = C endowed with the action of U H given by ∀ u ∈ U , ρ σ ( u ) = ε ( u ) Id σ and ρ σ ( H i ) = (cid:96)λ ( H i ) Id σ . Then the vector v (cid:48) = v ⊗ ∈ V (cid:48) = V ⊗ σ satifies ρ V (cid:48) ( x )( v (cid:48) ) (cid:54) = 0. Indeed, v (cid:48) has weight α (cid:48)(cid:48) and for y = x ( α (cid:48)(cid:48) ) = F α ( x )( α (cid:48)(cid:48) ) = F α ( x )( α (cid:48) ) ∈ W ⊂ U one has x.v (cid:48) = y.v (cid:48) = ( y (1) .v ) ⊗ ( ε ( y (2) )1) = ( y.v ) ⊗ F α ( x ) .v ) ⊗ (cid:54) = 0 . Now consider ¯ α ∈ G such that C ¯ α and C H ¯ α are semi-simple. Then thefinite dimensional algebra U ¯ α is isomorphic to the product of algebraEnd C ( V i ) for representatives ( V i ) i of the isomorphism class of simple U ¯ α -modules. Lift one of these modules V to a simple module V H ∈ C H ¯ α then by semi-simplicity, V H is projective in C H ¯ α thus also in C H andso is its image V in C . (cid:3) Example 4.22.
Combining Examples 2.3, 2.7, 4.17, Corollary 4.16 andProposition 4.21 we can conclude that Axioms 1–5 are satisfied for thequantum group U associated to sl at any root of unity ξ = exp( iπ(cid:96) )with (cid:96) ≥ (cid:96) / ∈ Z . Moreover, Examples 2.4, 2.8, 4.18, Corollary4.16 and Proposition 4.21 imply that Axioms 1–5 are satisfied for thequantum group U associated to simple finite dimensional complex Liealgebra for any root of unity ξ = exp( iπ(cid:96) ) with (cid:96) odd.5. Topological invariant of bichrome graphs
This section does not require the results of Section 4. It contains theconstruction of topological invariant corresponding from the algebra ofSection 3. In particular, in this section we use the topological ribbonHopf algebra (cid:99) U H and the category C H of weight modules over U H toconstruct a topological universal invariant. The invariant is defined onbichrome graphs colored by both the algebra (cid:99) U H and objects of C H .Every manifold we will consider in this paper will be oriented, everydiffeomorphism of manifolds will be positive, and every link and tanglewill be oriented and framed.Let L : U H (cid:98) ⊗ n → End C ( (cid:99) U H (cid:98) ⊗ n ) be the left multiplication: L x ( y ) = xy .5.1. Bichrome graphs.
For a non-negative integer n , we describe thecategory [ n ] C H as follows. The objects of [ n ] C H are vector spaces of ODIFIED GRADED HENNINGS INVARIANT 31 the form [ n ] V := ( U H ) (cid:98) ⊗ n ⊗ V for some object V of C H . A morphism in [ n ] C H from [ n ] V to [ n ] V (cid:48) is a morphism of topological (cid:99) U H -modules f : [ n ] V → [ n ] V (cid:48) which is ofthe form(34) f = ( L ξ q + l ⊗ Id) m (cid:88) i =1 L u i ⊗ f i for some u i ∈ U ⊗ n , a quadratic element q , a linear element l and forsome linear maps f i ∈ Hom C ( V, V (cid:48) ) , ≤ i ≤ m .Let f : [ n ] V → [ n ] V (cid:48) be a non-zero morphism as in Equation (34).Then f determines q and l ∈ ( G (cid:48) ) n the class of l modulo (Λ ∗ ) n uniquely.The pair ( q, l ) is called the exponent of f .As we will explain, bichrome ribbon graph mimic Turaev’s defini-tion of ribbon graphs, given in [34], but have additional informationincluding red and blue edges and special coupons. For more on rib-bon graphs, ribbon categories and their associated Reshetikhin-Turaevfunctors see [34]. HereAn n -string link is an ( n, n )-tangle whose i -th incoming boundaryvertex is connected to the i -th outgoing boundary vertex by an edgedirected from bottom to top for every 1 ≤ i ≤ n .A bichrome graph is a ribbon graph with edges divided into twogroups, red and blue, satisfying the following condition: for everycoupon there exists a number k ≥ k input legsand the first k output legs are red with positive orientation, meaningincoming and outgoing respectively, while all the other ones are blue.The smoothing of a coupon with k red edges is the union of k parallelred strings union the coupon with its red edges removed, see Figure1. Note this is a different smoothing than given in [10] where the bluepart was removed in this process.A n -string link graph is a bichrome ribbon graph Γ in R × [0 , n incoming boundaryvertices and the first n outgoing boundary vertices are red, while all theother ones are blue and, (2) the red sub-tangle of the graph obtainedby smoothing all coupons of Γ is an n -string link. We say such a graphis ( (cid:99) U H , C H )-colored if: (1) the red edges are colored by (cid:99) U H , (2) theblue edges colored by objects of C H and (3) coupons are colored bymorphisms in [ k ] C H (here k can vary for the different coupons).To simplify notation, when it is clear we will say n -string link graphfor a ( (cid:99) U H , C H )-colored n -string link graph. We also consider the specialkind of coloring on a n -string link graph: We say a n -string link graphis a Vect C -graph with (cid:99) U H -beads if: (1) the red edges are colored by (cid:99) U H , (2) the blue edges colored by objects of C H , (3) coupons do nothave any red edges and are colored by morphisms in Vect C and (4) (cid:32) Figure 1.
Smoothing of a couponthe graph is equipped with beads which form a finite set b of points onedges colored by an element of ( (cid:99) U H ) (cid:98) ⊗ b . Here and in what follows, if S is a finite set with n element the tensor product V ⊗ S means V ⊗ n wherethe factors are indexed by elements of S .5.2. The universal invariant.
Let U = HH (cid:16) (cid:99) U H (cid:17) := (cid:99) U H / [ (cid:99) U H , (cid:99) U H ]seen as a topological (cid:99) U H -module with trivial action, and lettr u : (cid:99) U H → U be the canonical projection.Let Γ be a n -string link graph. We will now define the universalinvariant associate to Γ which is an element J (Γ) ∈ ( U ) (cid:98) ⊗ C (cid:98) ⊗ ( U H ) (cid:98) ⊗ n ⊗ Hom C ( V, V (cid:48) )where C is the set of closed red components of Γ. Here V (respec-tively V (cid:48) ) is the tensor product of the colors of the blue bottom (re-spectively top) boundary points of Γ. We will also show that J (Γ) is (cid:99) U H -equivariant.Choose a diagram D of Γ which is a regular projection of Γ. Wedefine an element J ( D ) in the following three steps. Step 1, Construction of B ( D ) : Using the rules of Figure 2 we putbeads at the cup, the cap and the crossing (this is same as the usualuniversal invariant). In this figure the symbol S ↓ ( x ) for x ∈ (cid:99) U H is ODIFIED GRADED HENNINGS INVARIANT 33 S ↓ ( a ) S ↓ ( b ) S ↓ ( a ) S ↓ ( S − ( b )) 1 1 g g − Figure 2.
Place beads on the strings where R = (cid:80) a ⊗ b (cid:80) L x i ⊗ · · · ⊗ L x in ⊗ f i · · · V (cid:48) V (cid:39) (cid:80) x i · · · x i n f i V (cid:48) V Figure 3.
Smoothing a mixed coupon using beads.defined by S ↓ ( x ) = x if the arrow is upward , S ( x ) if the arrow is downward . Then we replace each coupon with k red edges with its smoothingequipped with k beads as shown in Figure 3. The result is a linearcombination B ( D ) of Vect C -graphs with (cid:99) U H -beads. Step 2, Construction of J bD via contraction: For each closed redcomponent of D fix a base point b . Using the rules of Figure 4, wereplace each bead on a blue strand colored by V by using the action of (cid:99) U H on V or its transpose . We obtain J bD ∈ ( U H ) (cid:98) ⊗ C (cid:98) ⊗ ( U H ) (cid:98) ⊗ n ⊗ Hom C ( V, V (cid:48) )by multiplying the beads on each red strand in order opposite to theorientation of the component (if the red component is closed start atthe fixed base point) and we separately evaluate the blue part of thegraph using the Penrose graphical calculus for Vect C -colored ribbongraphs. Step 3, Definition of J ( D ) via trace: For each closed red componentapply the trace tr u : (cid:99) U H → U to obtain the element J ( D ) = (cid:0) tr u ⊗ C ⊗ Id ⊗ n ⊗ Id (cid:1)(cid:0) J bD (cid:1) ∈ ( U ) (cid:98) ⊗ C (cid:98) ⊗ ( U H ) (cid:98) ⊗ n ⊗ Hom C ( V, V (cid:48) ) . Remark that the convention for beads on blue shown in Figure 4 is differentfrom [10] where we used t ρ V ( S ( x )). xV = ρ V ( x ) VV , xV = t ρ V ( x ) VV Figure 4.
Beads on blue edges are Vect C -colored coupons. (cid:80) L x i ⊗ f i fV V V V B −→ f i fV V V V x i a t b t a j b j a k S − ( b k )1 a s S ( b s ) g − Figure 5.
Putting beads on a graph D The element J ( D ) is independent of the choice of base points b becausethe starting point of multiplying the beads on any closed red circlecomponent does not matter as the result is sent in the quotient U .This construction illustrated for the diagram D given in Figure 5,where J ( D ) is an element of U (cid:98) ⊗ (cid:99) U H ⊗ Hom C ( V , V ) given by J ( D ) = (cid:88) tr u (cid:0) g − S ( b s )1 a k (cid:1) ⊗ x i b t a j S − ( b k ) a s ⊗ F C ( T )where F C ( T ) = (cid:16) Id V ⊗ ←− ev V (cid:17) ◦ ( f i ◦ ρ V ( a t b j ) ⊗ Id V ) ◦ f ∈ Hom C ( V , V ) . Here F C is the Reshetikhin-Turaev functor from the category of Vect C -colored ribbons graphs to the category of finite dimensional vectorspaces Vect C .The following theorem is an adapted version of the universal invari-ant of [20, 27, 22]: Theorem 5.1.
The element J ( D ) computed above on does not dependon the choice of D and is an isotopy invariant J (Γ) of Γ in R × [0 , . ODIFIED GRADED HENNINGS INVARIANT 35
Furthermore J (Γ) is (cid:99) U H -equivariant, i.e., (35) J (Γ) (cid:0) u (1) ⊗ · · · ⊗ u ( k ) ⊗ ρ V ( u ( k +1) ) (cid:1) = (cid:0) u (1) ⊗ · · · ⊗ u ( k ) ⊗ ρ V (cid:48) ( u ( k +1) ) (cid:1) J (Γ) for all u ∈ (cid:99) U H where ∆ k +1 ( u ) = u (1) ⊗ · · · ⊗ u ( k +1) and k equals n plusthe number of closed red components of Γ .Proof. We will use the symbol ≡ to relate two Vect C -graphs with (cid:99) U H -beads whose images are equal under F C .The following local relation for Vect C -graphs with (cid:99) U H -beads applyfor any color of their edges:(36) xy ≡ xy and xy ≡ yx . Also beads can freely move around cap and cup with any orientation:(37) x ≡ x and x ≡ x . Using these relations the invariance of J (Γ) under Redeimeister’s movesfollows the proof of [27] using the properties of the R-matrix. Theinvariance for the local move of a strand sliding over (or under) acoupon follows from the (cid:99) U H -equivariance of the morphisms coloringthe coupon. We will prove the invariance of the local move given inFigure 6, the other moves are proved similarly. After putting the beadsand smoothing the coupon the associated equality is (cid:88) a j a i ⊗ xb i ⊗ f ◦ b j = (cid:88) a t a s ⊗ b s x ⊗ b t ◦ f. This equality is equivalent to(38) (cid:16) ⊗ (cid:88) x ⊗ f (cid:17) R R = R R (cid:16) ⊗ (cid:88) x ⊗ f (cid:17) . We consider the right hand side of Equality (38), one gets R R (cid:16) ⊗ (cid:88) x ⊗ f (cid:17) = (Id ⊗ ∆)( R ) (cid:16) ⊗ (cid:88) x ⊗ f (cid:17) = (cid:16)(cid:88) a i ⊗ ∆( b i ) (cid:17)(cid:16) ⊗ (cid:88) x ⊗ f (cid:17) = (cid:88) a i ⊗ ∆( b i ) (cid:16)(cid:88) x ⊗ f (cid:17) = (cid:88) a i ⊗ (cid:16)(cid:88) x ⊗ f (cid:17) ∆( b i )= (cid:16) ⊗ (cid:88) x ⊗ f (cid:17) (Id ⊗ ∆)( R )= (cid:16) ⊗ (cid:88) x ⊗ f (cid:17) R R . In the third equality we used (cid:80) L x ⊗ f being (cid:99) U H -invariant. HenceEquality (38) holds. (cid:80) L x ⊗ f ∼ (cid:80) L x ⊗ f Figure 6.
Moving at local couponAs claimed above, the element J ( D ) is independant of the choice ofbase point: indeed, if one move the base point so that it pass a beadcolored by a ∈ U H the two corresponding values for J ( D ) will be thesame except for a factor which will be tr u ( xa ) in one case and tr u ( ax )on the other case but these two elements are equal in U so moving thebase does not change J ( D ).We now prove that J (Γ) is (cid:99) U H -equivariant, i.e., Equation (35) holds.One can compute the left hand side of Equation (35) by applying F C to the diagram B ( D ) where some beads colored by the u ( i ) have beenthreaded at the intersection point of D with an horizontal line near thebottom of D . Similarly, the right hand side is the image by F C of thediagram B ( D ) with some u ( i ) beads along a horizontal line near the topof D . For a horizontal line at any level (not intersecting coupons) wecan define similarly a diagrams obtain from B ( D ) with some S ↓ ( u ( i ) )beads at the intersection point of Γ with the horizontal line.We claim that all diagrams obtained by this process will have thesame image by F C . Indeed, it is enough to see this is true when onemoves the horizontal line to pass an elementary diagram. Except forred caps and cups, this follows from the (cid:99) U H -equivariance of the braid-ing given by τ R , of the morphisms coloring coupons and of the (co)-evalutations in C H . For red cups and caps it is a direct computation ;we develop here the case of red caps (see Figure 7): Assume the hori-zontal line under the cap intersect Γ at N + 2 points and at N pointswhen the line is above the cap. If the cap is joining the i th and the i +1 th strands, we can write ∆ N u = u (1) ⊗· · ·⊗ u ( i − ⊗ a ⊗ u ( i +1) ⊗· · ·⊗ u ( N +1) .Then ∆ N − u = u (1) ⊗ · · · ⊗ u ( i − ⊗ ε ( a ) ⊗ u ( i +1) ⊗ · · · ⊗ u ( N +1) and∆ N +1 u = u (1) ⊗ · · · ⊗ u ( i − ⊗ a (1) ⊗ a (2) ⊗ u ( i +1) ⊗ · · · ⊗ u ( N +1) . Thenthe statements for the red caps (see Figure 7) hold from the equalities S ( a (1) ) a (2) = ε ( a ) and S ( a (2) ) ga (1) = gS − ( a (2) ) a (1) = ε ( a ) g . The simi-lar statements for the red cups hold from the first equality above and a (2) g − S ( a (1) ) = ε ( a ) g − for a ∈ (cid:99) U H . (cid:3) ODIFIED GRADED HENNINGS INVARIANT 37 S ( a (1) ) a (2) ≡ , ε ( a ) a (1) g S ( a (2) ) ≡ g ε ( a ) Figure 7.
Equivariance for red caps
Lemma 5.2.
Let (cid:99) U H be the weight subspace of (cid:99) U H . Then U ∼ = (cid:99) U H / ([ U , (cid:99) U H ] ∩ (cid:99) U H ) . Proof.
First (cid:99) U H is a weight module and if x is homogeneous of degree λ (cid:54) = 0, then there exists h ∈ H such that λ ( h ) = 1 and then x = [ h, x ] isa commutator. Hence only weight 0 elements contribute to U . Next,remark that if x = f ( h ) w ∈ (cid:99) U H and y ∈ (cid:99) U H have opposite weights,then f ( h ) wy = wyf ( h ) and[ x, y ] = [ f ( h ) w, y ] = f ( h ) wy − yf ( h ) w = [ w, yf ( h )] ∈ [ U , (cid:99) U H ] . (cid:3) Value of the invariant.
Here we describe some properties ofthe invariant J of the previous section. Let D be a planar diagramrepresenting a n -string graph Γ in R × [0 ,
1] having m closed red com-ponents. In the rest of the paper we will assume that all n -string graphwill be homogenous : meaning that each blue edge is colored homoge-nous object of C H , see (7). In particular, we use the following notation:the j th blue edge is associated with an element ¯ β j ∈ G such that thisedge is colored by an object of C H ¯ β j .Choose a base point b for each closed red component of the diagram D . Consider the element J bD and Vect C -graph with (cid:99) U H -beads B ( D )given in Section 5.2. The diagram B ( D ) consists of m + n red compo-nents and a blue graph D bl with beads and n D bl edges. We numberedthe components and edges of D starting with the m red closed com-ponents then the n ordered open red components and finally the blueedges. As in Section 5.2 we express J (Γ) = (cid:0) tr u ⊗ m ⊗ Id ⊗ n ⊗ Id (cid:1)(cid:0) J bD (cid:1) ∈ U (cid:98) ⊗ m (cid:98) ⊗ ( U H ) (cid:98) ⊗ n ⊗ Hom C ( V, V (cid:48) )where J bD = (cid:80) w b ⊗ · · · ⊗ w b m ⊗ y ⊗ · · · ⊗ y n ⊗ F C ( D bl ) depends onthe choice b .The element J bD can be seen as a vector-valued entire function with r ( m + n ) variables: indeed, we can associate to the i th red component,the set of variables h i = ( H ,i , . . . , H r,i ) . Recall that a morphism f : [ k ] V → [ k ] V (cid:48) determines a pair ( q, l ) calledthe exponent determined by Equation (34). For each coupon c of D with k red legs let ( q, l ) be the exponent corresponding to the morphismin c with ( ξ q , l ) ∈ Q k × ( G (cid:48) ) k . For such a coupon, let i , i , . . . , i k ∈{ , . . . , m + n } be the labels (with possible repetition) of its adjacentred edges and let q ( c, D ) = q ( h i , . . . , h i k ) then ξ q ( c,D ) ∈ Q m + n and l ( c, D ) = l ( h i , . . . , h i k ) ∈ ( G (cid:48) ) m + n . Finally, let(39) Q cD = (cid:88) c coupon q ( c, D ) and L cD = (cid:88) c coupon l ( c, D ) . Next, we will use the elements Q cD and L cD to assign an exponentto J bD . This exponent is a sum of three contributions: the first is aquadratic element coming from the crossing between red strands, thesecond is a linear element coming from bichrome crossings and the lastis coming from the coupons.Let lk = (lk ij ) ≤ i,j ≤ m + n + n Dbl be the linking matrix associated D .Remark that the first ( m + n ) × ( m + n ) block of lk does not dependon the planar projection but the second does: the coefficient ( i, j ) isthe sum of the sign of crossings of the planar diagram where the strand j is above the strand i (see [6]). Recall the element Q = Q ( h , h ) = (cid:80) i c i ⊗ c (cid:48) i ∈ Λ ∗ ⊗ Z Λ ∗ ⊂ H ⊗ H which corresponds to the symmetricbilinear form B on Λ and H = ξ Q ( h ,h ) . Let Q D = (cid:88) ≤ k,s ≤ m + n lk ks Q ( h k , h s ) , so ξ Q D ∈ Q m + n , (40) L D = (cid:88) ≤ k ≤ m + n
Let L D and L cD be any representatives in H ( m + n ) of L D and L cD in ( G (cid:48) ) m + n = ( H / Λ ∗ ) m + n , respectively and let P D = Q D + Q cD + L D + L cD . Then the element ξ P D is independent of the choice ofthese representatives up to a unit in U ⊗ m + n (i.e. ξ to an integral linearcombination of the H i,j ) and (42) J bD ∈ ξ P D U ⊗ m + n ⊗ Hom C ( V, V (cid:48) ) . Proof.
The first statement of the theorem follows from Remark 3.8. Toprove Equation (42) notice that the elements H i,j form a commutativealgebra and by Lemma 3.9 power quadratic and power linear elementscommutes with U so all contributions to the exponent can be added inany order. We just have to check that the contribution of crossings isgiven by the elements Q D and L D .At each crossing point of type ( k, s ) in the diagram of B ( D ) withbeads for 1 ≤ k, s ≤ m + n , i.e., a red-red type crossing point, theCartan part of the R -matrix gives us the factor ξ ε Q ( h k ,h s ) ∈ Q m + n ODIFIED GRADED HENNINGS INVARIANT 39 where ε ∈ {± } is the sign of the crossing. Summing all these contri-butions, we get a factor ξ lk ks Q ( h k ,h s ) ∈ Q m + n . Next we consider a crossing point ( k, s ) in the the diagram of B ( D )for 1 ≤ k ≤ m + n, m + n < s ≤ m + n + n D bl , i.e., a red-blue typecrossing point. Suppose that the color of the blue strand is V ∈ C H ¯ β for ¯ β ∈ G . Then any weight λ of V is congruent to ¯ β modulo Λ andthe Cartan part of the element R -matrix acting on the λ weight spaceof V gives us a factor (see Remark 3.8) ξ ε ks Q ( λ,h k ) ∈ ξ ε ks Q ( ¯ β,h k ) U ⊗ m + n ⊂ L m + n U ⊗ m + n . So the k th red component and the s th blue component give the factor ξ (lk ks + lk sk ) Q ( ¯ β,h k ) ∈ L m + n U ⊗ m + n . (cid:3) Lemma 5.4.
Let D be as above and let ← D be the same diagram withthe orientation of the i th closed red component reversed ( ≤ i ≤ m ).Here we assume that this component does not pass through a coupon.Then J b ← D is obtained from J bD by applying the antipode or its inverse tothe i th factor, depending if the strand is oriented to the top or to thebottom at the base point.Proof. We can split the i th component into a sequence of segmentswhose end points are the extremal points (critical points of the secondcoordinate) p , . . . , p k at cups and caps (cups and caps alternate whenone follows the component).Let us assume first that the base point is on the segment [ p k , p ]which is oriented upward. Then multiplying together the beads oneach segment, the contribution to the i th factor of J bD has the form X = S ↓ ( a k ) g − ε k S ↓ ( a k − ) g ε k − S ↓ ( a k − ) · · · g − ε S ↓ ( a ) g ε S ↓ ( a )= a k g − ε k S ( a k − ) g ε k − a k − · · · g − ε S ( a ) g ε a where (see also Figure (2))(1) S ↓ ( a j ) is the product of the beads on the j th segment [ p j , p j +1 ]where the a j are products of factors of R or R − ,(2) the j th segment [ p j , p j +1 ] is oriented upward if j is even anddownward if j is odd,(3) ε j is 0 if the cap/cup is oriented to the left and 1 if it is orientedto the right.Then the corresponding contribution to the i th factor of J b ← D has theform X (cid:48) = S ( a ) g ε (cid:48) a g − ε (cid:48) · · · S ( a k − ) g ε (cid:48) k − a k − g − ε (cid:48) k S ( a k ) where ε (cid:48) j = 1 − ε j because a left oriented cap/cup become a rightoriented cap/cup (once again see Figure (2)). Then the lemma followsfrom the fact that for each jS (cid:0) g − ε j S ( a j − ) g ε j − (cid:1) = g − ε j − S ( a j − ) g ε j = g ε (cid:48) j − a j − g − ε (cid:48) j , so we have X (cid:48) = S ( X ).Since changing the orientation twice recover the initial diagram, and X = S − ( X (cid:48) ), we get that in case the base point is on a downwardsegment, the new invariant is obtained by applying S − . (cid:3) The following proposition established the universal character of thered colored strands:
Proposition 5.5.
Let Γ be a bichrome graph and Γ V be obtained byreplacing the i th colored red strand by a blue strand colored by V ∈ C H then (1) if the strand is closed, J (Γ V ) is obtained from J (Γ) by applying tr V C ◦ ρ V on the factor of U coresponding to this strand. (2) if the strand is the last open red strand, then J (Γ V ) is obtainedfrom J Γ by applying ρ V on the last factor of U H .Proof. This is a consequence from the fact that beads can be collectedwithout paying attention to the color of the strand by using the rela-tions (36) and (37). Then Penrose graphical calculus in Vect C does notdepends of the embedding of a strand because the braiding in Vect C istrivial so we can assume that the strand is either a simple circle or avertical string. (cid:3) Invariant of -manifolds The goal of this section is to define an invariant of the compatibletriples ( M, Γ , ω ) where, loosely speaking, M is a 3-manifold, Γ is aclosed bichrome graph inside M and ω ∈ H ( M \ Γ , G ). The mainingredients of the construction are the universal invariant defined inprevious section, the discrete Fourier transforms, the graded integral,the modified trace and the modified integral.6.1. Compatible triple.
A closed bichrome ribbon graph is a bichromeribbon graph with no boundary vertices. Let Γ be a n -string link graphin M = R × [0 ,
1] or a closed bichrome graph embedded in any ori-ented closed 3-manifold M . Let (cid:101) Γ be the smoothing of Γ in M . Let ω ∈ H ( M \ (cid:101) Γ , G ). We now defined the notion of a compatible triple,first in a simple situation and then the general case. First, we con-sider the situation with the following two requirements: 1) the bilinearform B (involve in the braiding) is non degenerate and 2) every closedred component of Γ is a framed knot with no coupons. With theserequirements we say the triple ( M, Γ , ω ) is compatible if: ODIFIED GRADED HENNINGS INVARIANT 41 (1) For each blue edge e of Γ colored by V ∈ C Hg with orientedmeridian m e , one has g = ω ( m e ).(2) For each i th closed red component of Γ with oriented parallel (cid:96) i , one has ω ( (cid:96) i ) = 0 ∈ G .For the general case, consider the element (cid:126)ω m ∈ G m + n obtained byevaluation of ω on the meridians of the m + n red components of (cid:101) Γ.By Lemma 3.7, the quadratic element Q c Γ of Equation (39) producesan element (cid:101) Q c Γ ( (cid:126)ω m , h • ) ∈ ( G (cid:48) ) m + n . Let f ci ∈ G (cid:48) be the i th componentof (cid:101) Q c Γ ( (cid:126)ω m , h • ) + L c Γ . Similarly, for Q ( h , h ) the exponant of H andfor any ¯ α ∈ G , we consider the element Q ( ¯ α, h ) ∈ G (cid:48) . Explicitly, if Q = (cid:80) i c i ( h ) ⊗ c (cid:48) i ( h ), then Q ( ¯ α, h ) = (cid:80) i c i ( ¯ α ) c (cid:48) i where the c i ( ¯ α ) ∈ C / Z are complex numbers modulo integers. Then we say the triple ( M, Γ , ω )is compatible if:(1) For each blue edge e of Γ colored by V ∈ C H ¯ β with orientedmeridian m e , one has ¯ β = ω ( m e ).(2) For each i th red closed component of Γ with oriented parallel (cid:96) i , one has 2 Q ( ω ( (cid:96) i ) , h ) + f ci = 0 ∈ G (cid:48) .If ( M, Γ , ω ) is a compatible triple where M is a closed 3-manifoldpresented by surgery on a link L ⊂ S then S \ ( (cid:101) Γ ∪ L ) ⊂ M \ (cid:101) Γ and inwhat follows we will still denote ω as the restriction of the cohomologyclass ω in H ( S \ ( (cid:101) Γ ∪ L ) , G ). Proposition 6.1.
Let ( M, Γ , ω ) be a compatible triple where M is a bean oriented closed 3-manifold presented by surgery on a link L ⊂ S .Then ( S , Γ ∪ L, ω ) is also a compatible triple where the components of L are all red.Proof. Condition (1) of a compatible triple is still satisfied because theblue edges Γ and Γ ∪ L are the same. We need to check condition (2)for the red components of L . The surgery link L has no coupon and itsparallels bound discs in M \ (cid:101) Γ. Hence for each parallel (cid:96) i of a componentof L , ω ( (cid:96) i ) = 0. Hence the triple ( S , Γ ∪ L, ω ) is compatible. (cid:3)
Kirby’s Theorem (given in [23]) loosely says that two surgery pre-sentations of a manifold can be connected by a sequence of isotopiesand Kirby moves. Consider the Kirby move given in Figure 8. Noticethat the Kirby 0 and II moves only occur on closed red componentswithout coupon.
Theorem 6.2.
Let ( M, Γ , ω ) and ( M (cid:48) , Γ (cid:48) , ω (cid:48) ) ) be compatible triple where M and M (cid:48) are oriented closed 3-manifolds. Let M and M (cid:48) be presentedby surgery on the links L and L (cid:48) , respectively. Suppose f : M → M (cid:48) is an be an orientation-preserving diffeomorphism such that f (Γ) = Γ (cid:48) as framed graph and f ∗ ( ω (cid:48) ) = ω . Then there exists a finite sequenceof isotopies, Kirby 0, Kirby I, Kirby II moves (see Figure 8) relating ¯ α ←→ − ¯ α ←→ ∅ ←→ Kirby 0 move Kirby I moves ¯ α ¯ β ←→ ¯ α ¯ β − ¯ α Kirby II move (black edge can be blue or red)The label on edges is the value of ω on their oriented meridian. Figure 8.
Colored Kirby moves( S , Γ ∪ L, ω ) and ( S , Γ (cid:48) ∪ L (cid:48) , ω (cid:48) ) such that the induced isomorphismis isotopic to f .Proof. In [30], the Kirby theorem is extended to the case of manifoldscontaining graphs. It is easy to see that the condition f ∗ ( ω (cid:48) ) = ω implies that this extension give a sequence of moves relating ( S , Γ ∪ L, ω ) and ( S , Γ (cid:48) ∪ L (cid:48) , ω (cid:48) ). Note a similar result was considered inTheorem 5.2 of [6]. (cid:3) Invariant of -manifolds. Let Γ be a n -string link graph with m red closed components. Let D be a planar diagram representing Γwith a choice of base point b for each closed red component. Recall weconstructed in Step 2 of Subsection 5.2 the element J bD ∈ ( U H ) (cid:98) ⊗ m + n ⊗ Hom C ( V, V (cid:48) ) (cid:39) C ω ( h • ) ⊗ W ⊗ m + n ⊗ Hom C ( V, V (cid:48) ) . So J bD is a vector-valued entire function with set of variables h • = { H i,j } i =1 ,...,r, j =1 ,...,m + n which are a basis of H ( m + n ) . Lemma 6.3.
Suppose Γ is equipped with a compatible cohomology class ω . Let −→ α = ( ¯ α , . . . , ¯ α m + n ) be the values of ω on the meridians ofthe red components. Then J bD is (cid:96) -periodic in the first rm variables { H i,j } i =1 ,...,r, j =1 ,...,m on Λ −→ α .Proof. Theorem 5.3 implies J bD ∈ ξ P D U ⊗ m + n ⊗ Hom C ( V, V (cid:48) ) ⊂ Q m + n L m + n U ⊗ m + n ⊗ Hom C ( V, V (cid:48) )where P D = Q D + Q cD + L D + L cD ∈ S ( H ( m + n ) ). Let us denote M = R × [0 , ω ∈ H ( M \ Γ; G )implies that the function ξ P D is (cid:96) -periodic in the first rm variables onΛ −→ α . ODIFIED GRADED HENNINGS INVARIANT 43
We check the periodicity of J bD ( h , . . . , h m + n ) in the set of variables h i = ( H ,i , . . . , H r,i ) for 1 ≤ i ≤ m by computing the evaluation of ξ P D at α + (cid:96)λ i with α ≡ −→ α mod Λ m + n and λ i = (0 , . . . , λ, . . . , ∈ ( H ∗ ) m + n ( λ at the i th position) for some λ ∈ Λ. In the following we used thepolarisation of quadratic element defined in Equation (23). One has P D ( α + (cid:96)λ i )= Q D ( α + (cid:96)λ i ) + Q cD ( α + (cid:96)λ i )+ L D ( α + (cid:96)λ i ) + L cD ( α + (cid:96)λ i )= P D ( α ) + Q D ( (cid:96)λ i ) + Q cD ( (cid:96)λ i ) + (cid:96) (cid:101) Q D ( α, λ i )+ (cid:96) (cid:101) Q cD ( α, λ i ) + (cid:96)L D ( λ i ) + (cid:96)L cD ( λ i )= P D ( α ) + Q D ( (cid:96)λ i ) + Q cD ( (cid:96)λ i ) + X, where X mod (cid:96) Z is given by¯ X = (cid:101) Q D ( (cid:96) −→ α , λ i ) + (cid:101) Q cD ( (cid:96) −→ α , λ i ) + (cid:96)L D ( λ i ) + (cid:96)L cD ( λ i ) . It is clear that Q D ( (cid:96)λ i ) + Q cD ( (cid:96)λ i ) = 0 mod (cid:96) Z . Now recall that Q = Q ( h , h ) is also symmetric and linear in each set of variable thus (cid:101) Q (( h , h ) , ( h (cid:48) , h (cid:48) )) = Q ( h + h (cid:48) , h + h (cid:48) ) − Q ( h , h ) − Q ( h (cid:48) , h (cid:48) )= Q ( h , h (cid:48) ) + Q ( h , h (cid:48) )then from (40), we get (cid:101) Q D ( (cid:96) −→ α , λ i ) = (cid:88) ≤ k,s ≤ m + n lk ks (cid:0) Q ( (cid:96) ¯ α k , δ si λ ) + Q ( (cid:96) ¯ α s , δ ki λ ) (cid:1) and (cid:96)L D ( λ i ) = (cid:88) ≤ k ≤ m + n
Let Γ be a n -string link graph with m red closedcomponents in R × [0 , equipped with a compatible cohomology class ω . Let −→ α = ( ¯ α , . . . , ¯ α m + n ) be the values of ω on the meridians of thered components. Let F µ (Γ , ω ) = ( µ ¯ α ⊗ · · · ⊗ µ ¯ α m ⊗ Id) (cid:0) J b Γ ,ω (cid:1) ∈ LU ¯ α m +1 ⊗ · · · ⊗ LU ¯ α m + n ⊗ Hom C ( V, V (cid:48) ) . Then F µ (Γ , ω ) does not depend on the base points, and is invariant byKirby 0 and Kirby II moves.Proof. Let us call L the sub-link of Γ formed by the m red closedcomponents. If one moves the base point of the i th component of L then the universal invariant J b Γ only change by terms which have acommutator c i at the i th factor. If such a term has non zero weight,then so will be its representant c i ∈ U ¯ α i and then µ ¯ α i ( c i ) = 0 by ODIFIED GRADED HENNINGS INVARIANT 45 • x ←→ • x (1) • x (2) • gB applied on a pair of diagrams related by a Kirby II move Figure 9.
Invariance by Kirby II moveEquation (32). Now if the weight of c i is zero, it can be written asthe commutator of two elements in LU ¯ α i . From the proof of Lemma5.2 we see that c i belongs to [ U ¯ α i , LU ¯ α i ] ∩ U ¯ α i = [ U ¯ α i , U ¯ α i ]. Then, fromEquation (30) we see that µ ¯ α i ( c i ) = 0. Hence F µ (Γ , ω ) does not dependof the base points.We now prove F µ (Γ , ω ) does not change under orientation reversalof components of L (i.e., the Kirby 0 move). First, by Lemma 5.4,changing the orientation of a component of L changes J b Γ by applying anantipode on the factor corresponding to the component, so the propertyof G -trace in Equation (31) imply F µ (Γ , ω ) does not depend on theorientation.Finally, to see the Kirby II move hold, we consider a red component K ⊂ L with beads with product x ∈ (cid:99) U H and a blue or red verticaledge e with locally no bead or equivalently a bead 1, as in the firstcomponent of Figure 9. Here the base point of K is assumed to benear the vertical edge and the orientation of K as in the figure. Let( ω (cid:48) , −→ α (cid:48) ) be the image of ( ω, −→ α ) by the second Kirby move. After themove, a pivotal element g has appeared on the slidding edge e and sincethe two curves are parallel with the same orientation, all the elementsalong the component K are replaced by x (1) on K and x (2) on e where∆ x = x (1) ⊗ x (2) . Since modulo I −→ α , x ∈ U ¯ β we have that modulo I −→ α (cid:48) ,∆ x ∈ U ¯ β − ¯ α ⊗ U ¯ α . Then (28) implies that the values of F µ before andafter the Kirby II move are the same. (cid:3) Up to isotopy, one can identify closed bichrome graphs in R × [0 , S . The same holds for pair (Γ , ω )where ω is a compatible cohomology class in the complement of thesmoothing (cid:101) Γ of a closed bichrome graph Γ. Hence Proposition 6.4induce a complex valued invariant of compatible triple ( S , Γ , ω ) thatwe still denote by F µ (Γ , ω ) ∈ C . Theorem 6.5 (Graded Hennings invariant) . Let ( M, Γ , ω ) be compat-ible triple where M is a oriented closed 3-manifolds. Let L ⊂ S bea framed link which is a surgery presentation of M . Denote ω as therestriction of the cohomology class ω in H ( S \ ( (cid:101) Γ ∪ L ) , G ) . Define (43) H( M, Γ , ω ) = δ − s F µ ( L ∪ Γ , ω ) where δ = µ ( g − θ ) and s is the signature of the linking matrix of L .Then H is a well defined invariant of diffeomorphism class of ( M, Γ , ω ) .Proof. From Proposition 6.1, we have ( S , Γ ∪ L, ω ) is a compatibletriple and by Theorem 6.2 any two such presentation are related byKirby moves. By Proposition 6.4, F µ ( L ∪ Γ , ω ) is invariant by Kirby0 and Kirby II moves. For the Kirby I move, if we add a ± L with meridian m , then ω ( m ) = 0 because the class of m is zero in H ( M \ (cid:101) Γ , Z ). Then the invariant F (cid:48) is multiplied by λ ( θ ) ∓ = δ ∓ . At the same time the signature of the linking matrix s changes by ±
1, so H( M, Γ , ω ) does not change under the Kirby Imove. (cid:3) In many known examples the invariant H is zero for “generic” com-patible triples. In particular, in the case of Example 2.3, the invariantH is zero when Γ is colored with a projective module or when ω hasnon-integral values. However, as we will now explain, m-traces can beused to renormalize H to define a non-zero invariant for these “generic”compatible triples. Proposition 6.6.
Let Γ be a n -string link graph and ( R × [0 , , Γ , ω ) bea compatible triple, then F µ (Γ , ω ) ∈ LU ¯ α ⊗· · ·⊗LU ¯ α n ⊗ Hom C ( V, V (cid:48) ) isequivariant: write F µ (Γ , ω ) = (cid:80) i x i ⊗ f i then let U H ⊗−→ α = U H ¯ α (cid:98) ⊗ · · · (cid:98) ⊗U H ¯ α n where the factors are equipped with the action of (cid:99) U H by left multiplica-tion. Then (cid:88) i L x i ⊗ f i ∈ Hom C (cid:0) U H ⊗−→ α ⊗ V, U H ⊗−→ α ⊗ V (cid:48) (cid:1) is a morphism of (cid:99) U H -modules.Proof. The proof is by induction on the number m of closed red com-ponents of Γ. If m = 0, by Equation (35) of Theorem 5.1, the universalinvariant J (ˇΓ) is (cid:99) U H -equivariant. Since the projection modulo theideals I ¯ α is equivariant, then F µ (Γ , ω ) is (cid:99) U H -equivariant.Now assume m > n +1)-string link graph ˇΓ whoseleft braid closure of the first strand produces Γ. Let ˇ D be a diagram ofˇΓ and D its left braid closure obtained by joining the m top endpointsof ˇ D to its m bottom endpoints using an arc disjoint from the diagramon its left. We still denotes by ω the restriction of the cohomology classto R × [0 , \ ˇΓ ⊂ R × [0 , \ Γ. Then by induction F µ (ˇΓ , ω ) is (cid:99) U H -equivariant. Now computing the invariants using ˇ D and D with the ODIFIED GRADED HENNINGS INVARIANT 47 new base point on the left arc, we have that if F µ (ˇΓ , ω ) = (cid:80) i y i ⊗ z i ⊗ f i with y i ∈ U ¯ α then(44) F µ (Γ , ω ) = (cid:88) i µ ¯ α ( y i g − ) z i ⊗ f i . Now for u ∈ U , let ∆ ( n +2) u = u a ⊗ u b ⊗ u c with u a ∈ U , u b ∈ U ⊗ n and u c ∈ U . We can write ∆ n +1 u = ε ( u a ) u b ⊗ u c and then F µ (Γ , ω )(∆ n +1 u ) = (cid:88) i µ ¯ α ( y i ε ( u a ) g − ) z i u b ⊗ f i ( u c · )= (cid:88) i µ ¯ α ( y i u a (2) S − ( u a (1) ) g − ) z i u b ⊗ f i ( u c · )= (cid:88) i µ ¯ α ( u a (2) y i g − S ( u a (1) )) u b z i ⊗ u c f i = (cid:88) i µ ¯ α ( S ( u a (1) ) u a (2) y i g − ) u b z i ⊗ u c f i = (cid:88) i ε ( u a ) µ ¯ α ( y i g − ) u b z i ⊗ u c f i = (∆ n +1 u ) F µ (Γ , ω ) , where the third equality uses the equivariance of F µ (ˇΓ , ω ) and thecyclicity of the symmetrised integral (30). For the H -linearity, we needto use Equation (32). By this equality only the terms of (44) where y i commute with any element H of H will contribute to the sum. Butsince H is primitive, ∆ ( n +2) H = H ⊗ ⊗ ∆ ( n +1) H so projectingEquation (44) on weight zero space for the first factor, we have (cid:88) | y i | =0 y i ⊗ (cid:0) ( z i ⊗ f i )∆ ( n +1) H (cid:1) = (cid:88) | y i | =0 y i ⊗ (cid:0) ∆ ( n +1) H ( z i ⊗ f i ) (cid:1) which implies the H linearity of F µ (Γ , ω ). (cid:3) Definition 6.7.
The compatible triple ( M, Γ , ω ) is called a graph-admissible triple if there exists a blue edge of Γ colored by V ∈ Proj ( C H ).Let ( S , Γ , ω ) be a graph-admissible triple with Γ is a closed bichromegraph embedded in S . A cutting presentation of Γ is a bichromegraph Γ V ∈ R × [0 ,
1] with bottom and top end point ( V, +) whosebraid closure is isotopic to ( S , Γ). It is equipped with the inducedcohomology class still denoted ω . Then Proposition 6.6 implies that F µ (Γ V , ω ) ∈ Hom C ( V, V ) is a morphism of (cid:99) U H -modules (as n = 0).But V is finite dimensional and so F µ (Γ V , ω ) is a morphism of U H -modules, i.e. F µ (Γ V , ω ) ∈ End C H ( V ). Let(45) F (cid:48) µ (Γ , ω ) = t V ( F µ (Γ V , ω )) . Proposition 6.8.
The modified invariant F (cid:48) µ (Γ , ω ) of Equation (45) does not depend on the choice of a cutting presentation Γ V . Further-more, it is invariant by Kirby 0 and Kirby II move and for any closed compatible triple ( S , Γ (cid:48) , ω (cid:48) ) , the modified invariant of the disjoint unionis given by F (cid:48) µ (Γ (cid:116) Γ (cid:48) , ω (cid:48)(cid:48) ) = F (cid:48) µ (Γ , ω ) F µ (Γ (cid:48) , ω (cid:48) ) where ω (cid:48)(cid:48) is the cohomology class induced by ω and ω (cid:48) .Proof. The proof of the first statement is similar to that in [14]: indeed,if Γ V and Γ V (cid:48) are both cutting presentation of Γ then there exists Γ V,V (cid:48) such that Γ V and Γ V (cid:48) are the partial braid closure of Γ V,V (cid:48) , and Γ
V,V (cid:48) conjugated by the braiding, respectively. Then we have t V ( F µ (Γ V , ω )) = t V (ptr V (cid:48) ( F µ (Γ V,V (cid:48) , ω )))= t V ⊗ V (cid:48) ( F µ (Γ V,V (cid:48) , ω ))= t V (cid:48) ⊗ V (cid:0) c − V (cid:48) ,V F µ (Γ V,V (cid:48) , ω ) c V (cid:48) ,V (cid:1) = t V (cid:48) (cid:0) ptr V (cid:0) c − V (cid:48) ,V F µ (Γ V,V (cid:48) , ω ) c V (cid:48) ,V (cid:1)(cid:1) = t V (cid:48) ( F µ (Γ V (cid:48) , ω ))where the second and fourth equality (resp. third equality) follow fromthe partial trace property (resp. cyclicity) of the m-trace. Now theinvariance by Kirby 0 and 2 moves of F (cid:48) µ (Γ , ω ) follows from that of F µ (Γ V , ω ). Finally for a disjoint union, just remark that is Γ V is acutting presentation of Γ, then Γ V (cid:116) Γ (cid:48) is a cutting presentation ofΓ (cid:116) Γ (cid:48) and F µ (Γ V (cid:116) Γ (cid:48) , ω (cid:48)(cid:48) ) = F µ (Γ V , ω ) F µ (Γ (cid:48) , ω (cid:48) ). (cid:3) Then the modified invariant naturally extends to graph-admissibletriples ( M, Γ , ω ) with Γ a closed bichrome graph embedded in M : Theorem 6.9 (Modified graded Hennings invariant) . Let M be an ori-ented closed -manifold and ( M, Γ , ω ) be a graph-admissible compatibletriple with surgery presentation ( S , Γ ∪ L, ω ) . Define (46) H (cid:48) ( M, Γ , ω ) = δ − s F (cid:48) µ ( L ∪ Γ , ω ) where δ = µ ( g − θ ) and s is the signature of the linking matrix of L .Then H (cid:48) is a well defined invariant of diffeomorphism class of ( M, Γ , ω ) .Furthermore, if ( M (cid:48) , Γ (cid:48) , ω (cid:48) ) is any closed compatible triple, the modifiedinvariant of the connected sum is H (cid:48) ( M M (cid:48) , Γ (cid:116) Γ (cid:48) , ω (cid:48)(cid:48) ) = H (cid:48) ( M, Γ , ω )H( M (cid:48) , Γ (cid:48) , ω (cid:48) ) where ω (cid:48)(cid:48) is the cohomology class induced by ω and ω (cid:48) .Proof. From Theorem 6.2 it is enough to show that H (cid:48) is invariant underthe Kirby moves. Proposition 6.8 imply the Kirby 0 and Kirby II hold.As in the proof of Theorem 6.5, if we add a ± L with meridian m , then ω ( m ) = 0 and the invariant F (cid:48) is multipliedby λ ( θ ) ∓ = δ ∓ . So the Kirby I move holds. The last propertyof the theorem, follows from considering a surgery presentation of theconnected sum, which is a disjoint union of surgery presentations in S . (cid:3) ODIFIED GRADED HENNINGS INVARIANT 49 Modified symmetrized integral
We introduce the notion of a modified integral which allows us torelax the admissibility condition for the modified graded Hennings in-variant and in particular gives an invariant for empty 3-manifolds.Given ¯ α i ∈ G , let −→ α = ( ¯ α , . . . , ¯ α n ), ¯ β j = (cid:80) ni = j ¯ α i for j ≤ n and¯ β = ¯ β . Denote U ⊗−→ α = U ¯ α ⊗ U ¯ α ⊗ ... ⊗ U ¯ α n and let ∆ −→ α : U ¯ β → U ⊗−→ α be the map∆ −→ α = (1 ⊗ ... ⊗ ⊗ ∆ ¯ α n − , ¯ β n ) ... (1 ⊗ ∆ ¯ α , ¯ β )∆ ¯ α , ¯ β . By definition of a Hopf G -coalgebra ∆ −→ α is an algebra morphism. Wedenote by Z −→ α the centralizer of ∆ −→ α ( U ¯ β ) which is a subalgebra of U ⊗−→ α formed by elements which commute with any element of ∆ −→ α ( U ¯ β ). Inparticular, for n = 1 and −→ α = ( ¯ α ) then Z ¯ α is the center of U ¯ α . Lemma 7.1.
Recall L g is the left multiplication of g . We have ( µ ¯ α L g − α ⊗ Id)( Z (¯ α, ¯ β ) ) ⊂ Z ¯ β and (Id ⊗ µ ¯ β L g ¯ β )( Z (¯ α, ¯ β ) ) ⊂ Z ¯ α . Proof.
Let z = (cid:80) z ⊗ z ∈ Z (¯ α, ¯ β ) , u ∈ U ¯ β and let ∆ ( − ¯ α, ¯ α, ¯ β ) ( u ) = u (1) ⊗ u (2) ⊗ u (3) . Then (cid:80) u (1) ⊗ z u (2) ⊗ z u (3) = (cid:80) u (1) ⊗ u (2) z ⊗ u (3) z by definition of Z (¯ α, ¯ β ) so (cid:88) µ ¯ α ( g − α z ) z u = (cid:88) µ ¯ α ( g − α z u (2) S − − ¯ α ( u (1) )) z u (3) = (cid:88) µ ¯ α ( g − α S − ¯ α ( u (1) ) z u (2) ) ⊗ z u (3) = (cid:88) µ ¯ α ( g − α S − ¯ α ( u (1) ) u (2) z ) ⊗ u (3) z = (cid:88) µ ¯ α ( g − α z ) uz . The proof is similar for the second inclusion. (cid:3)
Definition 7.2.
Let X be a subset of G . We say that a family ofmaps { µ (cid:48) ¯ α : Z ¯ α → C } ¯ α ∈ G \ X is a modified integral on G \ X if for any¯ α, ¯ β ∈ G \ X the following two linear forms are equal on Z (¯ α, ¯ β ) : µ ¯ α L g − α ⊗ µ (cid:48) ¯ β = µ (cid:48) ¯ α ⊗ µ ¯ β L g ¯ β . Theorem 7.3.
Let X be the subset of G such that G \ X is the setof ¯ α where U ¯ α is semi-simple. Then there exists a family of centralelements { z ¯ α ∈ Z ¯ α } ¯ α ∈ G \ X such that µ ¯ α ( x ) = tr C U ¯ α ( L z ¯ α L x ) for all x ∈U ¯ α . Furthermore, there exists a modified integral on G \ X defined by (47) µ (cid:48) ¯ α ( z ) := µ ¯ α ( z ¯ α z ) = tr C U ¯ α ( L z α z ) for all z ∈ Z ¯ α .Proof. Since U ¯ α is a finite-dimensional semi-simple algebra for ¯ α ∈ G \ X then it is isomorphic to a product of matrix algebras: U ¯ α ∼ = (cid:76) i Mat( V i ) where V i = C n i are its irreducible representations. Thenthe identity matrices of each summand form a basis { z i } i of the center Z α and the characters x (cid:55)→ tr C U ¯ α ( L z i L x ) = n i tr C V i ( ρ V i ( x )) form a basis ofHH ( U ¯ α ) ∗ := ( U ¯ α / [ U ¯ α , U ¯ α ]) ∗ . Finally since µ ¯ α ∈ HH ( U ¯ α ) ∗ there exists { δ i ∈ C } i such that µ ¯ α ( x ) = (cid:80) i δ i tr C U ¯ α ( L z i L x ) for all x ∈ U ¯ α . We define z ¯ α = (cid:80) i δ i z i .Remark that since z i z j = δ ji z i , applying the above formula for x = z i gives µ ¯ α ( z i ) = tr C U ¯ α ( L z i ) δ i = n i δ i because L z i is the identity of the blockMat( V i ). But the symmetrized integral of z i is the modified trace of theinduced endomorphism of U ¯ α (by left or right multiplication) which is aprojector on V ⊕ n i i then µ ¯ α ( z i ) = n i d i where d i is the modified dimensionof V i (see [18]), thus δ i = d i n i . Finally, one gets(48) µ ¯ α = (cid:88) i d i tr C V i ◦ ρ V i . Let us now prove that µ (cid:48) ¯ α = µ ¯ α ( z ¯ α · ) is a modified integral. Let x ∈ Z ¯ α, ¯ β and consider the endomorphism f of U ¯ α ⊗U ¯ β given by left multiplicationby ( z ¯ α ⊗ z ¯ β ) x . Then the properties of the partial trace imply that(49) t U ¯ α (ptr U ¯ β ( f )) = t U ¯ β (ptr U ¯ α ( f ))now the partial trace in the category is the partial trace in Vect twistedby the action of the pivotal elements and the modified trace on U ¯ α isgiven by t U ¯ α = µ ¯ α = tr C U ¯ α ( L z ¯ α · ) thus Equation (49) becomestr C U ¯ α ⊗U ¯ β ( L z α ⊗ g ¯ β z ¯ β L x ) = tr C U ¯ α ⊗U ¯ β ( L z ¯ α g − α ⊗ z β L x )with tr C U ¯ α ⊗U ¯ β = tr C U ¯ α ⊗ tr C U ¯ β so µ (cid:48) ¯ α ⊗ µ ¯ β ((1 ⊗ g ¯ β ) x ) = µ ¯ α ⊗ µ (cid:48) ¯ β (( g − α ⊗ x )which proves that µ (cid:48) is a modified integral. (cid:3) Remark that the modified integral µ (cid:48) ¯ α of Theorem 7.3 has a naturalextension as a linear form on the whole algebra U ¯ α . We call µ (cid:48) ¯ α the canonical modified integral.Recall the definition of a graph-admissible triple given in Defini-tion 6.7. Definition 7.4.
A compatible triple ( M, Γ , ω ) is called a G -admissibletriple if there exists a red edge of Γ colored by ¯ α ∈ G \ X . A triple( M, Γ , ω ) is called an admissible triple if ( M, Γ , ω ) is a G -admissibletriple or a graph-admissible triple.As for graph-admissible closed bichrome graph, we can introduce thenotion of cutting presentation Γ ¯ α of G -admissible triple ( S , Γ , ω ) bycutting a red edge rather than a blue edge: Γ ¯ α is a 1-string link graphwhose braid closure is Γ and where ω take on the meridian of the openstrand the value ¯ α ∈ G \ X . Remark that the universal invariant of sucha graph is (cid:96) -periodic in its r -variables and equivariant so F µ (Γ ¯ α ) ∈ Z ¯ α . Proposition 7.5.
There exists unique extensions of F (cid:48) µ to G -admissiblebichrome closed graphs (resp. H (cid:48) to G -admissible triples) ) with F (cid:48) µ (Γ , ω ) = µ (cid:48) ¯ α ( F µ (Γ ¯ α , ω )) ODIFIED GRADED HENNINGS INVARIANT 51 Γ V = T , Γ ¯ α = T Figure 10.
Cutting presentation Γ V at blue edge andΓ ¯ α at red edge where Γ ¯ α is a cutting presentation of Γ . The extension of H (cid:48) is givenby H (cid:48) ( M, Γ , ω ) = δ − s F (cid:48) µ ( L ∪ Γ , ω ) where ( S , Γ ∪ L, ω ) is a surgery presentation of an admissible triple ( M, Γ , ω ) , the constant δ is µ ( g − θ ) and s is the signature of thelinking matrix of L .Proof. Suppose that Γ admits two G -admissible cutting presentationsΓ ¯ α and Γ ¯ β . Then cutting the two edges we get a 2-string link graphwhose image by F µ is in Z ( α,β ) because of Proposition 6.6. Then Def-inition 7.2 implies that the invariant computed by Γ ¯ α and Γ ¯ β are thesame.Suppose now that Γ admits a G -admissible cutting presentation Γ ¯ α obtained by cutting a red edge and a graph-admissible cutting presen-tation Γ V for some projective V ∈ C H ¯ β obtained by cutting a blue edge.Then cutting the two edges lead to a graph T whose image by F µ isin U ¯ α ⊗ End C ( V ) (see Figure 10). Let us write F µ ( T, ω ) = L x ⊗ f (weomit the sum). Then µ (cid:48) α ( F µ (Γ ¯ α , ω )) = µ (cid:48) α L x ⊗ f = µ (cid:48) α ( x ) tr C U β (cid:0) ρ V ( g β ) f (cid:1) = µ α ( z α x ) tr C H R ( f ) = t U α ( L z α L x ) tr C U β (cid:0) ρ V ( g β ) f (cid:1) = F (cid:48) µ L x ⊗ fz α = t V L x ⊗ fz α = tr C U α (cid:16) L z α L g − α L x (cid:17) t V ( f ) = µ α ( xg − α ) t V ( f )= t V L x ⊗ f = t V ( F µ (Γ V , ω )) where the third equality follows because x tr C H R ( f ) ∈ Z ¯ α is centraland so tr C H R ( f ) L z ¯ α x = tr C H R ( f ) R z ¯ α x has its modified trace given byEquation (33). (cid:3) Relations with other non semi-simple invariants
We compare the invariant of this paper with three previously definedinvariants: (1) the one from the second author with sl (2 | The sl (2 | case. Let (cid:96) ≥ U ξ sl (2 |
1) andunrolled super algebra U Hξ sl (2 |
1) associated with the super Lie algebra sl (2 | (cid:99) U Hξ sl (2 |
1) is a topological ribbon Hopf superalgebra and its bosonization (cid:99) U Hξ sl (2 | σ = (cid:99) U Hξ sl (2 | (cid:111) Z / Z is a topo-logical ribbon Hopf algebra. Consider the bosonization U ξ sl (2 | σ ofthe semi-restricted super quantum group. This algebra can be realizedas the U of this paper where Λ is the root lattice of sl (2 |
1) and W is the Λ-graded vector space generated by the “E” and “F” parts ofthe Poincar´e–Birkhoff–Witt basis. It follows from [17] that U satisfiesthe five Axioms 1–5. Moreover, one can identify (cid:99) U Hξ sl (2 | σ with theunrolled version (cid:99) U H of U as defined in Section 3.The construction of [17] was the motivation for this paper. It pro-duces an invariant J ( M, ω ) which is equal to the graded Henningsinvariant of this paper: J ( M, ω ) = H( M, ∅ , ω ) . In addition, a special property for sl (2 |
1) is that δ = 1 (see [17, Lemma4.13]).8.2. Comparison with invariants from nilpotent weight mod-ules of unrolled quantum group.
In [6] an invariant of a compat-ible triple ( M, Γ , ω ) where Γ is a blue C H -ribbon graph embedded in M was defined from the data of a relative premodular G -category C H with translation group Z . Many examples of unrolled quantum groupslead to such categories and in particular quantum groups of Lie alge-bras at suitable root of unity. Let us discuss how relative premodularcategories fit into this paper.Let Λ be a lattice and W a Λ-graded vector space which give rise toan algebra U satisfying Axioms 1 and 2. One of the requirements of In [6] the term relative modular is used instead of relative premodular, later in[8] it was shown that an additional requirement is need to define a TQFT and sothe use of term was changed.
ODIFIED GRADED HENNINGS INVARIANT 53 a relative premodular category is the existence of a translation group.The lattice naturally gives a translation group, in the category of weightmodules, which is the family of one dimensional modules ( C λ ) λ ∈ Λ where w ∈ W acts by multiplication by ε ( w ) and H i by (cid:96)λ ( H i ). Here ourgroup G is H ∗ / Λ. Another requirement of relative premodular G -category is the (generic) semisimplicity of the C Hg for g ∈ G \ X with X being “small.” For the rest of the subsection, we assume that U satisfies Axioms 1–5 and that C H is a relative premodular G -category.We have proved this is the case for all semi-restricted quantum groupsassociated to simple Lie algebras.Let ¯ α ∈ G \ X so that U ¯ α is semisimple. Let Θ ¯ α be a set of simple U H ¯ α -module such that { ResF( V i ) : V i ∈ Θ ¯ α } is a set of representant of theisomorphism classes of simple U ¯ α -modules. Then Equation (48) impliesthat the character given by the Kirby color x (cid:55)→ (cid:80) V i ∈ Θ ¯ α d i tr C ( ρ V i ( x ))is equal on U ¯ α to the symmetrized integral µ ¯ α . Hence we get Proposition 8.1.
Let Γ be a monochrome blue C H -ribbon graph and ( M, Γ , ω ) a compatible triple. The invariant N C H ( M, Γ , ω ) , defined in [6] , coming from the relative premodular category C H is equal to themodified Hennings invariant H (cid:48) ( M, Γ , ω ) .Proof. As in [6], choose a computable presentation of ( M, Γ , ω ) thenone gets a diagram where the surgery components are colored by Kirbycolors of degree ¯ α / ∈ X . From Proposition 5.5, the invariant of this dia-gram is equal to the invariant F (cid:48) µ of the diagram obtained by replacingthe Kirby-colored component with red component. But this is preciselythe process for computing H (cid:48) . (cid:3) Comparison with the modified Hennings invariants.
Let U be an algebra as in Section 2 satisfying Axioms 1–5 and U H its unrolledversion. Recall that the quotient U H of U H contains a finite dimensionalribbon Hopf algebra U with the same R-matrix R ∈ Q U ⊗ /I (0 , (cid:39)U ⊗ U . There is an obvious associated ribbon functor C H → U -Mod.The construction of [10] produces a modified Hennings invariant sinceAxiom 4 ensure that U is a non-degenerate finite dimensional Hopfalgebra. Theorem 8.2.
Let ( M, Γ , ω ) be compatible triple with ω = 0 . Thenthe invariant H (cid:48) ( M, Γ , ω ) of this paper, associated to U , is equal to the(non graded) modified Hennings invariant of [10] associated to the finitedimensional ribbon Hopf algebra U .Proof. If ω = 0, then the definition of H (cid:48) only uses the integral µ = λ ( g · ) and the computation of the universal invariant can be done di-rectly in U as it is done in [10]. (cid:3) Remark that combining this theorem with Proposition 8.1 gives anew proof that the invariants of [6] and [10] coincide for zero cohomol-ogy classes, as was first shown in [9].
References [1] N. Andruskiewitsch and C. Schweigert. On unrolled Hopf algebras.
Journal ofKnot Theory and Its Ramifications , Vol. 27, No. 10, 1850053 (2018), 2018.[2] J. Barrett and B. Westbury. Spherical categories.
Adv. Math. , 143:357 – 375,1999.[3] A. Beliakova, C. Blanchet, and A. M. Gainutdinov. Modified trace is a sym-metrised integral. arXiv:1801.00321 , 2017.[4] H. Cartan. Vari´et´es analytiques complexes et cohomologie. (French) Colloquesur les fonctions de plusieurs variables, tenu `a Bruxelles , pages 41–55, 1953.[5] V. Chari and A. Pressley.
A guide to quantum groups . Cambrige UniversityPress, 1995.[6] F. Costantino, N. Geer, and B. Patureau-Mirand. Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories.
Jour-nal of Topology , pages 1005–1053, 2014.[7] F. Costantino, N. Geer, and B. Patureau-Mirand. Some remarks on the un-rolled quantum group of sl (2). J. Pure Appl. Algebra , pages 3238–3262, 2015.[8] M. De Renzi. Non-semisimple extended topological quantum field theories.
ToAppear in Mem. Amer. Math. Soc., arXiv:1703.07573 .[9] M. De Renzi, N. Geer, and B. Patureau-Mirand. Non-semisimple quantuminvariants and TQFTs from small and unrolled quantum groups. preprint toappear in Algebr. Geom. Topol. , arXiv:1812.10685, 2018.[10] M. De Renzi, N. Geer, and B. Patureau-Mirand. Renormalized Hennings in-variants and 2 + 1-TQFTs.
Commun. Math. Phys. , 362 (3):855 – 907, 2018.[11] N. Geer, J. Kujawa, and B. Patureau-Mirand. M-traces in (non-unimodular)pivotal categories. arXiv:1809.00499 , 2018.[12] N. Geer and B. Patureau-Mirand. Topological invariants from unrestrictedquantum groups.
Algebraic & Geometric Topology , 2013.[13] N. Geer and B. Patureau-Mirand. The trace on projective representations ofquantum groups.
Letters in Mathematical Physics , Volume 108, Issue 1:117–140, 2018.[14] N. Geer, B. Patureau-Mirand, and V. Turaev. Modified quantum dimensionsand re-normalized links invariants.
Compositio Mathematica , pages 196–212,2009.[15] N. Geer, B. Patureau-Mirand, and A. Virelizier. Traces on ideals in pivotalscategories.
Quantum Topology , 4, No. 1:91–124, 2013.[16] A. Grothendieck. R´esum´e des r´esultats essentiels dans la th´eorie des produitstensoriels topologiques et des espaces nucl´eaires.
Annales de l’Institut Fourrier ,tome 4:p. 73–112, 1952.[17] N. P. Ha. A Hennings type invariant of 3-manifolds from a topological Hopfsuperalgebra. arXiv:1806.08277 , pages 1–35, 2018.[18] N. P. Ha. Modified trace from pivotal Hopf G-coalgebras.
Journal of Pure andApplied Algebra , 224(5):106225, 2020.[19] I. Heckenberger. Lusztig isomorphisms for drinfel’d doubles of bosonizations ofnichols algebras of diagonal type.
Journal of Algebra , 323:2130 – 2182, 2010.[20] M. Hennings. Invariants of links and 3-manifolds obtained from Hopf algebras.
Journal of the London Mathematical Society , 54:594–624, 1996.[21] C. Kassel.
Quantum Groups . Springer-Verlag, 1995.
ODIFIED GRADED HENNINGS INVARIANT 55 [22] L. Kauffman and D. E. Radford. Oriented quantum algebras, categories andinvariants of knots and links.
Journal of Knot Theory and Its Ramifications ,10(07):1047–1084, 2001.[23] R. Kirby. A calculus for framed links.
Invent. Math. , 45:35–56, 1978.[24] S. Lentner and T. Ohrmann. Factorizable R -Matrices for Small QuantumGroups. SIGMA , 13, Number 076:1–25, 2017.[25] V. Lyubashenko. Invariants of 3-manifolds and projective representations ofmapping class groups via quantum groups at roots of unity.
Communicationsin Mathematical Physics , 172, Issue 3:467–516, September 1995.[26] T. Ohtsuki. Colored ribbon Hopf algebras and universal invariants of framedlinks.
J. Knot Theory Ramifications 2 (2) , pages 211–232, 1993.[27] T. Ohtsuki.
Quantum invariants . World Scientific Publishing Co. Pte. Ltd,2002.[28] M. J. Pflauma and M. Schottenloher. Holomorphic deformation of H opf alge-bras and applications to quantum groups. Journal of Geometry and Physics ,Volume 28, Issues 1-2:31–44, 1998.[29] D. E. Radford.
Hopf Algebras . World Scientific, 2012.[30] J. Roberts. Kirby calculus in manifolds with boundary.
Turkish J. Math. ,21:111–117, 1997.[31] F. Tr`eves.
Topological vector spaces, Distributions and Kernels , volume 25.Academic Press Inc., 1967.[32] V. Turaev. Crossed group-categories. preprint arXiv math/0005291 , 2000.[33] V. Turaev.
Homotopy Quantum Field Theory . European Mathematical Society,2010.[34] V. G. Turaev.
Quantum invariants of knots and 3-manifolds , volume 18 of
DeGruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, 1994.[35] A. Virelizier.
Alg`ebres de Hopf gradu´ees et fibr´es plats sur les 3-vari´et´es . Th`esede doctorat, Universit´e Louis Pasteur, 2001.[36] A. Virelizier. Hopf group-coalgebra.
Journal of Pure and Applied Algebra ,171:75–122, 2002.
Utah State University, Department of Mathematics and Statistics,Logan UT 84341, USA
E-mail address : [email protected] Hung Vuong University, Faculty of Natural Sciences, Viet Tri, PhuTho, Viet Nam
E-mail address : [email protected] Universit´e Bretagne Sud, Laboratoire de Math´ematiques de Bre-tagne Atlantique, UMR CNRS 6205, Campus de Tohannic, BP 573 F-56017 Vannes, France
E-mail address ::