aa r X i v : . [ m a t h . QA ] F e b ON JORDAN ALGEBRAS AND SOME UNIFICATION RESULTS
Florin F. Nichita
Simion Stoilow Institute of Mathematics of the Romanian Academy21 Calea Grivitei, 010702 Bucharest, Romania
Abstract
This paper is based on a talk given at the 14-th International Workshop on Differ-ential Geometry and Its Applications, hosted by the Petroleum Gas University fromPloiesti, between July 9-th and July 11-th, 2019. After presenting some historicalfacts, we will consider some geometry problems related to unification approaches.Jordan algebras and Lie algebras are the main non-associative structures. Attemptsto unify non-associative algebras and associative algebras led to UJLA structures.Another algebraic structure which unifies non-associative algebras and associative al-gebras is the Yang-Baxter equation. We will review topics relared to the Yang-Baxterequation and Yang-Baxter systems, with the goal to unify constructions from Differ-ential Geometry.
Keywords:
Jordan algebras, Lie algebras, associative algebras, Yang-Baxter equa-tions
MSC-class:
Introduction
This paper is a survey paper, but it also presents new results. It emerged after ourlectures given at the 9–th Congress of Romanian Mathematicians (CRM 9, Galati,Romania, June-July 2019) and 14-th International Workshop on Differential Geometryand Its Applications (DGA 14, UPG Ploiesti, July 2019).We begin with some historical remarks. Vr˘anceanu proposed the study of spaceswith constant affine connection associated to finite-dimensional real Jordan algebrasin 1966. This study was afterwards developed by Iordanescu, Popovici and Turtoi(see the Comments on page 31 from [14]). Dan Barbilian proved that the rings whichcan be the underlying rings for projective geometries are (with a few exceptions) ringswith a unit element in which any one-sided inverse is a two-sided inverse (see [14]).Another big Romanian geometer was G. Tzitzeica, and some of his contributions tomathematics will be presented in our next section.A substantial part of this paper is dedicated to unifying structures. The apparitionof the Yang-Baxter equation ([32]) in theoretical physics and statistical mechanics (see[40, 2, 3]) has led to many applications in these fields and in quantum groups, quantumcomputing, knot theory, braided categories, analysis of integrable systems, quantummechanics, etc (see [30]). The interest in this equation is growing, as new properties ofit are found, and its solutions are not classified yet. This equation can be understoodas an unifying equation (see, for example, [31, 20, 21, 15]). Another unification ofnon-associative structures, was obtained using the UJLA structures ([15, 16]), which ould be seen as structures which comprise the information encapsulated in associativealgebras, Lie algebras and Jordan algebras.We now refer to important results obtained under the guidance of St. Papadima.Some presentations at DGA 14 were dedicated to his memory (see [4, 34, 28, 37]).It was observed during DGA 14, that some results from [34] can be extended for thevirtual braid groups from [4]. Also, the techniques of [37] can be considered in theframework of UJLA structures. The Yang–Baxter equation plays an important role inknot theory. D˘asc˘alescu and Nichita have shown in [8] how to associate a Yang–Baxteroperator to any algebra structure over a vector space, using the associativity of themultiplication. Turaev has described in [39] a general scheme to derive an invariantof oriented links from a Yang–Baxter operator, provided this one can be “enhanced”.The invariant which was obtained from those Yang–Baxter operators is the Alexanderpolynomial of knots. Thus, in a way, the Alexander polynomial is the knot invariantcorresponding to the axioms of (unitary associative) algebras.The current paper is organised as follows. The next section deals with Tzitzeica-Johnson’s theorem. In Sections 3 and 4 we present the Yang-Baxter equations andthe Yang-Baxter systems(including the set-theoretical Yang-Baxter equation). Thesewill be followed by a section on Jordan algebras, UJLA structures and unificationconstructions in Differential Geometry. A conclusion section will end this article.2. On Tzitzeica-Johnson’s theorem and pictorial mathematics
In his talk at DGA 12, Florin Caragiu explained that there exists a special math-ematical discourse, called “proofs without words”, which uses pictures or diagramsin order to boost the intuition of the reader (see [7]). Pictorial (diagrammatic) styleof mathematical language is much appreciated by both educators and researchers inmathematics (see, also, [26]). Very easy to be grasped, some pictorial style problemsneed an entirely “artillery” in order to be cracked.We now consider the Tzitzeica-Johnson’s problem (see, for example, [26]). We startwith three circles of the same radius r . The intersection points of pairs of circles aredenoted by A, B, C and the common point of intersection of the three circles is denotedby O (see the Figure 1 below). Figure 1.
The three coins problem (Tzitzeica - 1908, Johnson -1916). Under the above assumptions, there exists a circle with radius r , passing through the points A, B and C. ✫✪✬✩ ✫✪✬✩✫✪✬✩✫✪✬✩ A O BC
We now propose a new construction (in the Figure 2). igure 2.
The tangent to the third circle in O meets the line BC in R. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✁✁✁✁✁✁✁✁ ✫✪✬✩✫✪✬✩✫✪✬✩
A O RBC
At this moment, we can present our
Theorem : under the above assumptions, thetangents to the circles in O meet the lines AB, AC and BC in three colinear points(see the Figure 3 below).
Figure 3.
Theorem.
The points P, Q and R are colinear. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)☞☞☞☞☞☞☞☞✥✥✥✥✥✥❅❅❅❅❅❅❅❅❅ ❚❚❚❚❚❚❚❚❚❚ ✫✪✬✩✫✪✬✩✫✪✬✩
Q PR
The proof of the above theorem is based on the properties of the power of a circleand on the Desargues’ theorem. It is the limit case of the following picture.
Figure 4.
The limit case of the following picture implies our theorem. ✱✱✱✱✱✱❧❧❧❧❧❧ ✫✪✬✩✫✪✬✩✫✪✬✩
The above problems can be included in a more general scheme (see the Figure 5).
Figure 5.
A mathematical scheme. If X , Y and Z have certainproperties, then there exists a related T , with the same properties. ✫✪✬✩✫✪✬✩✫✪✬✩ X ZY he dual of the Desargues’ Theorem can be interpretated in the light of the abovescheme by using arrangements of (three) colored lines.The hypothesis: X = { ( d, f, h ) such that d ∩ f ∩ h = P }Y = { ( d ′ , f, h ′ ) such that d ′ ∩ f ∩ h ′ = Q }Z = { ( d ′′ , f, h ′′ ) such that d ′′ ∩ f ∩ h ′′ = R }X ∩ Y ∩ Z = { f }X ∩ Y = { A, A ′ } ∪ { f }Y ∩ Z = { B, B ′ } ∪ { f }X ∩ Z = { C, C ′ } ∪ { f } Conclusion:There exist a triple ( l, m, n ) through
A, A ′ , B, B ′ , C, C ′ .From the phylosophical point of view, one could include in this scheme the inclussion-exclusion principle : | X ∪ Y ∪ Z | = | X | + | Y | + | Z | − | X ∩ Y | − | X ∩ Z | − | Y ∩ Z | + | X ∩ Y ∩ Z | .Let us explain this analogy. Once we have information about the cardinality of X, Y, Z, X ∩ Y, X ∩ Z, Y ∩ Z and X ∩ Y ∩ Z , then there exists a T , related to X, Y and Z , whose cardinality can be also computed. Of course, in this case T = X ∪ Y ∪ Z .In order to illustrate better this analogy, let us suppose further that | X | = | Y | = | Z | = 7, | X ∩ Y | = | X ∩ Z | = | Y ∩ Z | = 1 and | X ∩ Y ∩ Z | = 1. Then, there exists aset T , with | T | = 7 which contains elements from all possible (sub)sets.Notice that Tzitzeica-Johnson’s Theorem has many generalizations ([26]) and in-terpretations related to the above mathematical scheme (from Figure 5).Thus, Tzitzeica-Johnson’s Theorem can be interpretated in terms of disks:(i) “If three disks of the same radius have a common point of intersection, thenthey contain inside of their union a forth disk with the same radius.”(ii) “If three disks X , Y and Z of the same radius have a common point, then thereexists a forth disk of the same radius which includes ( X ∩ Y ) ∪ ( X ∩ Z ) ∪ ( Y ∩ Z ).”It is an open problem to prove a similar statement for a domain bounded by anarbitary closed convex curve. In other words we conjecture that if we consider adomain X bounded by an arbitary closed convex curve, and two copies of X , denotedby Y and Z such that X ∩ Y ∩ Z 6 = ∅ , then there exists another copy of X , denotedby T such that ( X ∩ Y ) ∪ ( X ∩ Z ) ∪ ( Y ∩ Z ) ⊂ T .We now consider another type of problems.In his best-seller book [10], P. Greco (see [33] - there is a puzzle there!) refers to aproblem which can be generalized as follows. OPEN PROBLEMS. For an arbitrary convex closed curve, we consider the lengthof largest diameter (D). Here, the largest diameter is the longest segment through thecenter of mass with the endpoints on the given curve. In a similar manner, one candefine the smallest diameter (d).i) If L is the length of the given curve and the domain inside the given curve is aconvex set, then we conjecture that: LD ≤ π ≤ Ld . ii) If A is the area inside the given curve, the equation (1) x − L x + A = 0 and its implications are not completely understood. For example, if the given curve isan ellipse, solving this equation in terms of the semi-axes of the ellipse is an unsolvedproblem. The second part of the above conjecture is related to [1].Another idea related to [10] and to the Euler’s relation, e πi + 1 = 0 , is thefollowing relation containing π : | e i − π | > e . It is very good approximation, and ithas a pictorial interpretation (see the Figure 6). Figure 6.
An interpretation for the formula | e i − π | ≈ e . ✟✟✟✟✟✟✟❍❍❍❍❍☞☞☞☞☞☞ A BC π b B ≈ radian Yang-Baxter equations
In the next sections, we will work over a generic field k . The tensor products willbe defined over k .This section was inspired by two papers, [25, 38], and it begins with an introductionto the Yang-Baxter equation.Let V be a vector space over k . Let I = I V : V → V be the identity map of thespace V .We denote by τ : V ⊗ V → V ⊗ V the twist map defined by τ ( v ⊗ w ) = w ⊗ v .Now, we will give the main notations for introducing the Yang-Baxter equation.For R : V ⊗ V → V ⊗ V a k -linear map, let R = R ⊗ I, R = I ⊗ R : V ⊗ V ⊗ V → V ⊗ V ⊗ V .
In a similar manner, we denote by R a linear map acting on the firstand third component of V ⊗ V ⊗ V . It turns out that R = ( I ⊗ τ )( R ⊗ I )( I ⊗ τ ). Definition 3.1. A Yang-Baxter operator is k -linear map R : V ⊗ V → V ⊗ V ,which satisfies the braid condition (the Yang-Baxter equation): (2) R ◦ R ◦ R = R ◦ R ◦ R . We also require that the map R is invertible. Remark 3.2.
Some examples of Yang-Baxter operators are the following: R = τ (i.e., R ( a ⊗ b ) = b ⊗ a ), and R = I ⊗ I (i.e., R ( a ⊗ b ) = a ⊗ b ). emark 3.3. An important observation is that if R satisfies (2) then both R ◦ τ and τ ◦ R satisfy the QYBE (the quantum Yang-Baxter equation): (3) R ◦ R ◦ R = R ◦ R ◦ R . Remark 3.4.
The equations (2) and (3) are equivalent.
Remark 3.5.
The following construction of Yang-Baxter operators is described in [8] .If A is a k -algebra, then for all non-zero r, s ∈ k , the linear map (4) R Ar,s : A ⊗ A → A ⊗ A, a ⊗ b sab ⊗ r ⊗ ab − sa ⊗ b is a Yang-Baxter operator.The inverse of R Ar,s is ( R Ar,s ) − ( a ⊗ b ) = r ab ⊗ s ⊗ ab − s a ⊗ b . Turaev has described in [39] a general scheme to derive an invariant of orientedlinks from a Yang–Baxter operator, provided this one can be “enhanced”.The Jones polynomial [18] and its two–variable extensions, namely the Homflyptpolynomial [9, 35] and the Kauffman polynomial [19], can be obtained in that wayby “enhancing” some Yang–Baxter operators obtained in [17]. Those solutions ofthe Yang–Baxter equation are associated to simple Lie algebras and their fundamen-tal representations. The Alexander polynomial can be derived from a Yang–Baxteroperator as well, using a slight modification of Turaev’s construction [27].The only invariant which can be obtained from the Yang–Baxter operators (4) isthe Alexander polynomial of knots. Thus, in a way, the Alexander polynomial is theknot invariant corresponding to the axioms of (unitary associative) algebras (cf. [24]).Note that specializations of the Homflypt polynomial had to be expected from thoseYang–Baxter operators since they have degree 2 minimal polynomials.
Remark 3.6.
There is a similar terminology for the set-theoretical Yang-Baxter equa-tion. In this case V is replaced by a set X and the tensor product by the Cartesianproduct. We will explain this definition in the next examples below. Let X be a set containing three logical sentences p, q, r (i.e., p, q, r ∈ X ). We canchoose X as rich as we wish for the moment. Later, we will try to find the smallest X which fits for our theory.Let R : X × X → X × X , be defined by R ( p, q ) = ( p ∨ q, p ∧ q ).It follows that(5) ( R × I ) ◦ ( I × R ) ◦ ( R × I ) = ( I × R ) ◦ ( R × I ) ◦ ( I × R ) . One way to check that (5) holds is to make a table with values for p, q, r . Remark 3.7.
Another interesting solution to the set-theoretical Yang-Baxter equationis the following.Let R : X × X → X × X , be defined by R ( p, q ) = ( p → q, p ) .Again, one way to check the above statement is to make a table with values for p, q, r . Let us denote R ( p, q ) = ( p , q ). The set-theoretical Yang-Baxter equation can beexpressed as6) ( p , q , r ) = ( p , q , r ) . It is now the moment to discuss about X . Open problems.
What can be said about X in general? What is the smallest X for which R is well-defined? One could consider a set X containing more thanthree logical sentences. What is the interpretation of the set-theoretical Yang-Baxtersolutions in this case ?We can go a step further and consider an algebra of type (2,2), ( A, ∗ , ◦ ), and call theoperations ∗ and ◦ YB conjugated if R ( a, b ) = ( a ∗ b, a ◦ b ) satisfies the set-theoreticalYang-Baxter equation.We propose the study of algebras with two operations which are YB conjugated.Groups, distributive lattices, and self-distributive laws can be considered as objectswith operations which are YB conjugated (compare with [5]). Remark 3.8.
We now consider other (systems of ) equations for a k -linear map R : V ⊗ V → V ⊗ V : (7) R ◦ R = R ◦ R = R ◦ R (8) R ◦ R = R ◦ R (9) R ◦ R ◦ R ◦ R = R ◦ R ◦ R ◦ R (10) ( R ◦ R ◦ R − R ◦ R ◦ R ) ◦ ( R ◦ R ◦ R − R ◦ R ◦ R ) = 0(11) R = XY , R ◦ X ◦ R ◦ Y = R ◦ X ◦ R ◦ Y (12) R ◦ R = I ⊗ I (13) R R R R + R R R R = R R R R R R + R R R R R R Theorem 3.9.
If a k -map R : V ⊗ V → V ⊗ V verifies (7) and (8) then R is acommon solution for (2) and (3).If a k -map R : V ⊗ V → V ⊗ V verifies (2) and (3) then it is a solution for (9).If a k -map R : V ⊗ V → V ⊗ V verifies (2) or (3) then R is a solution for (10).If a k -map R : V ⊗ V → V ⊗ V verifies (2) or (3) then it is a solution for (11). Proof.
We only prove the first claim: R ◦ R ◦ R = R ◦ R ◦ R = R ◦ R ◦ R R ◦ R ◦ R = R ◦ R ◦ R = R ◦ R ◦ R The other claims follow in a similar manner. ⋄ We will write the above results in the following manner:(7) ∧ (8) → (2) ∧ (3); (2) ∧ (3) → (9);(2) ∨ (3) → (10); (2) ∨ (3) → (11); (10) ∧ (12) → (13).As a direct application of this theorem, one can check which of the funtions pre-sented in this section are common solutions for the braid condition and the quantumYang-Baxter equation. emark 3.10. The set of equations of operators of type R : V ⊗ V → V ⊗ V and R ij : V ⊗ V ⊗ V → V ⊗ V ⊗ V , with i, j ∈ { , , } , has a natural distributive laticestructure (see [25] ). Yang-Baxter systems
We present the Yang-Baxter systems theory following the paper [6]. Yang-Baxtersystems were introduced in [11] as a spectral-parameter independent generalisation ofquantum Yang-Baxter equations related to non-ultralocal integrable systems.Yang-Baxter systems are conveniently defined in terms of
Yang-Baxter commuta-tors . Consider three vector spaces
V, V ′ , V ′′ and three linear maps R : V ⊗ V ′ → V ⊗ V ′ , S : V ⊗ V ′′ → V ⊗ V ′′ and T : V ′ ⊗ V ′′ → V ′ ⊗ V ′′ .Then a Yang-Baxter commutator is a map [
R, S, T ] : V ⊗ V ′ ⊗ V ′′ → V ⊗ V ′ ⊗ V ′′ ,defined by(14) [ R, S, T ] = R ◦ S ◦ T − T ◦ S ◦ R . In terms of a Yang-Baxter commutator, the quantum Yang-Baxter equation (3) isexpressed simply as [
R, R, R ] = 0.
Definition 4.1.
Let V and V ′ be vector spaces. A system of linear maps W : V ⊗ V → V ⊗ V, Z : V ′ ⊗ V ′ → V ′ ⊗ V ′ , X : V ⊗ V ′ → V ⊗ V ′ is called a WXZ-system or a
Yang-Baxter system , provided the following equationsare satisified: (15) [
W, W, W ] = 0 , (16) [ Z, Z, Z ] = 0 , (17) [ W, X, X ] = 0 , (18) [ X, X, Z ] = 0 . There are several algebraic origins and applications of WXZ-systems. WXZ-systemswith invertible W , X and Z can be used to construct dually-paired bialgebras of theFRT type, thus leading to quantum doubles.More precisely, consider a WXZ-system with finite-dimensional V = V ′ , so that eachof W , X , Z is an N × N -matrix. Suppose that W, X, Z are invertible. Since W and Z satisfy Yang-Baxter equations (15)–(16), one can consider two matrix bialgebras A and B with N × N matrices of generators U and T respectively, and relations W U U = U U W , Z T T = T T Z . The existence of an invertible operator X that satisfies equations (17)–(18), means that A and B are dually paired with anon-degenerate pairing < U , T > = X . Furthermore, the tensor product A ⊗ B hasan algebra (quantum double) structure with crossed relations X U T = T U X .Given a WXZ-system as in Definition 4.1 one can construct a Yang-Baxter operatoron V ⊕ V ′ , provided the map X is invertible. This is a special case of a gluingprocedure described in [23, Theorem 2.7] (cf. [23, Example 2.11]). Let R = W ◦ τ V,V , R ′ = Z ◦ τ V ′ ,V ′ , U = X ◦ τ V ′ ,V . Then the linear map R ⊕ U R ′ : ( V ⊕ V ′ ) ⊗ ( V ⊕ V ′ ) → ( V ⊕ V ′ ) ⊗ ( V ⊕ V ′ )iven by R ⊕ U R ′ | V ⊗ V = R , R ⊕ U R ′ | V ′ ⊗ V ′ = R ′ , and for all x ∈ V , y ∈ V ′ ,( R ⊕ U R ′ )( y ⊗ x ) = U ( y ⊗ x ) , ( R ⊕ U R ′ )( x ⊗ y ) = U − ( x ⊗ y )is a Yang-Baxter operator.Entwining structures were introduced in order to recapture the symmetry structureof non-commutative (coalgebra) principal bundles or coalgebra-Galois extensions. Definition 4.2.
An algebra A is said to be entwined with a coalgebra C if there existsa linear map ψ : C ⊗ A → A ⊗ C satisfying the following four conditions:(1) ψ ◦ ( I C ⊗ µ ) = ( µ ⊗ I C ) ◦ ( I A ⊗ ψ ) ◦ ( ψ ⊗ I A ) ,(2) ( I A ⊗ ∆) ◦ ψ = ( ψ ⊗ I C ) ◦ ( I C ⊗ ψ ) ◦ (∆ ⊗ I A ) ,(3) ψ ◦ ( I C ⊗ ι ) = ι ⊗ I C ,(4) ( I A ⊗ ε ) ◦ ψ = ε ⊗ I A .The map ψ is known as an entwining map , and the triple ( A, C ) ψ is called an entwining structure . Remark 4.3.
To denote the action of an entwining map ψ on elements it is conve-nient to use the following α -notation , for all a, b ∈ A and c ∈ C , ψ ( c ⊗ a ) = P α a α ⊗ c α , ( I A ⊗ ψ ) ◦ ( ψ ⊗ I A )( c ⊗ a ⊗ b ) = P α,β a α ⊗ b β ⊗ c αβ , etc.For example (6) is a kind of α -notation related to the Yang-Baxter equation.The relations (1), (2), (3) and (4) in Definition 4.2 are equivalent to the followingexplicit relations, for all a, b ∈ A , c ∈ C , (19) X α ( ab ) α ⊗ c α = X α,β a α b β ⊗ c αβ , (20) X α a α ⊗ c α (1) ⊗ c α (2) = X α,β a βα ⊗ c (1) α ⊗ c (2) β , (21) X α α ⊗ c α = 1 ⊗ c, (22) X α a α ε ( c α ) = aε ( c ) . Theorem 4.4. ( [6] ) Let A be an algebra and let C be a coalgebra. For any s, r, t, p ∈ k define linear maps W : A ⊗ A → A ⊗ A, a ⊗ b sba ⊗ r ⊗ ba − sb ⊗ a,Z : C ⊗ C → C ⊗ C, c ⊗ d tε ( c ) X d (1) ⊗ d (2) + pε ( d ) X c (1) ⊗ c (2) − pd ⊗ c. Let X : A ⊗ C → A ⊗ C be a linear map such that X ◦ ( ι ⊗ Id C ) = ι ⊗ Id C and ( Id A ⊗ ε ) ◦ X = Id A ⊗ ε . Then W, X, Z is a Yang-Baxter system if and only if A isentwined with C by the map ψ := X ◦ τ C,A . . Jordan algebras and UJLA structures
Jordan algebras have applications in physics, differential geometry, ring geometries,quantum groups, analysis, biology, etc (see [12, 14, 13]).We have introduced some structures which unify Jordan algebras, Lie algebras and(non-unital) associative algebras. These structures were called UJLA (from “unifi-cation”, “Jordan”, “Lie” and “associative”). structures, and one could “decode” theresults obtained for UJLA structures in results for Jordan algebras, Lie algebras or(non-unital) associative algebras.Changhing the perspective, one can consider the UJLA structures as generalizationsof Jordan algebras.
UJLA str . k − alg ✻ UJLA strJordan alg ✻✲✲ { , } I I { , } UJLA structures can also be interpretated as in an intermediat step in the processof associationg a Lie algebra to an associative algebra (see the picture below).
UJLA str . k − alg ✻ UJLA strk − Lie alg ❄✲✲ [ , ] I [ , ][ , ]The study of filiform Lie algebras ([36]) can be extended to filiform UJLA structures. Definition 5.1.
We have defined the unifying structure ( V, η ) , also called a “UJLAstructure”, in the following way. Let V be a vector space, and η : V ⊗ V → V, η ( a ⊗ b ) = ab, be a linear map, which satisfies the following axioms ∀ a, b, c ∈ V : (23) ( ab ) c + ( bc ) a + ( ca ) b = a ( bc ) + b ( ca ) + c ( ab ) , (24) ( a b ) a = a ( ba ) , (25) ( ab ) a = a ( ba ) , (26) ( ba ) a = ( ba ) a , (27) a ( ab ) = a ( a b ) . If just the identity (23) holds, we call the structure ( V, η ) a “weak unifying struc-ture”. emark 5.2. If ( A, θ ) , where θ : A ⊗ A → A, θ ( a ⊗ b ) = ab , is a (non-unital)associative algebra, then we define a UJLA structure ( A, θ ′ ) , where θ ′ ( a ⊗ b ) = αab + βba , for some α, β ∈ k . For α = and β = , then ( A, θ ′ ) is a Jordan algebra, andfor α = 1 and β = − , then ( A, θ ′ ) is a Lie algebra. Theorem 5.3.
Let ( V, η ) be a UJLA structure, and α, β ∈ k . Then, ( V, η ′ ) , η ′ ( a ⊗ b ) = αab + βba is a UJLA structure. Remark 5.4.
Let ( V, η ) be a UJLA structure. Then, δ a : V → V, δ a ( x ) = xa − ax ,is a derivation for the following UJLA structure: ( V, η ′ ) , η ′ ( a ⊗ b ) = ab − ba . OPEN PROBLEM.
A UJLA structure is power associative. We know that aUJLA structure is power associative for dimensions less or equal to 5.
Remark 5.5.
The classification of UJLA structures is also an open problem.
Theorem 5.6.
Let V be a vector space over the field k , and p, q ∈ k . For f, g : V → V , we define M ( f ⊗ g ) = f ∗ g = f ∗ p,q g = pf ◦ g + qg ◦ f : V → V . Then:(i) ( End k ( V ) , ∗ p,q ) is a UJLA structure ∀ p, q ∈ k .(ii) For φ : End k ( V ) → End k ( V ⊗ V ) a morphism of UJLA structures (i.e., φ ( f ∗ g ) = φ ( f ) ∗ φ ( g ) ), W = { f : V → V | f ◦ M = M ◦ φ ( f ) } is a sub-UJLA structure ofthe structure defined at (i). In other words, f ∗ g ∈ W, ∀ f, g ∈ W . Theorem 5.7.
Let ( V, η ) be a UJLA structure. Then, ( V, η ′ ) , η ′ ( a ⊗ b ) = ab − ba isa Lie algebra. Proof.
The proof follows from formula (23). (cid:3)
Theorem 5.8.
Let ( V, η ) be a UJLA structure. Then, ( V, η ′ ) , η ′ ( a ⊗ b ) = ( ab + ba ) is a Jordan algebra. Proof.
The proof follows from formulas (24), (25), (26) and (27). (cid:3)
Theorem 5.9.
For ( V, η ) a UJLA structure, D ( x ) = D b ( x ) = bx − xb is a UJLA–derivation (i.e., D ( a a ) = D ( a ) a + a D ( a ) ∀ a ∈ V ). Proof.
In formula (23) we take c = a : ( ab ) a + ( ba ) a + ( a a ) b = a ( ba ) + b ( a a ) + a ( ab ). It follows that ( ba ) a + ( a a ) b = b ( a a ) + a ( ab ).So, ( ba ) a − a ( ab ) = b ( a a ) − ( a a ) b ; so, b ( a a ) − ( a a ) b = ( ba − a b ) a + a ( ba − ab ).Thus, D ( a a ) = D ( a ) a + a D ( a ). Definition 5.10.
For the vector space V, let d : V → V and φ : V ⊗ V → V ⊗ V, bea linear map which satisfies: (28) φ ◦ φ ◦ φ = φ ◦ φ ◦ φ where φ = φ ⊗ I, φ = I ⊗ φ , I : V → V, a a .Then, ( V, d, φ ) is called a generalized derivation if φ ◦ ( d ⊗ I + I ⊗ d ) = ( d ⊗ I + I ⊗ d ) ◦ φ . Remark 5.11. If A is an associtive algebra, d : A → A a derivation (so, d (1 A ) =0),and φ : A ⊗ A → A ⊗ A, a ⊗ b ab ⊗ ⊗ ab − a ⊗ b , then ( A, d, φ ) is a generalizedderivation.f C is a coalgebra, d : C → C a coderivation, and ψ : C ⊗ C → C ⊗ C, c ⊗ d ε ( d ) c ⊗ c + ε ( c ) d ⊗ d − c ⊗ d , then ( C, d, ψ ) is a generalized derivation.If τ is the twist map, the condition τ ◦ R ◦ τ = R represents the unification of the comutativity and the co-comutativity conditions. Inother words, if the algebra A is comutative, then φ verifies the above condition. If thecoalgebra C is cocomutative, then ψ verifies the same condition. Definition 5.12.
Let A is an associtive algebra, d : A → A a derivation, M anA-bimodule, and D : M → M with the property D ( am ) = d ( a ) m + aD ( m ) . Then, ( A, d, M, D ) is called a module derivation. Theorem 5.13. ( [29] ) In the above case, A × M becomes an algebra, and δ : A × M → A × M, ( a, m ) ( d ( a ) , D ( m )) is a derivation in this algebra. Translated into the “language” of Differential Geometry, the above theorem saysthat the Lie derivative is a derivation (i.e., d ( ab ) = d ( a ) b + ad ( b ) ) on the product ofthe algebra of functions defined on the manifold M with the set of vector fields on M(see [22]). Remark 5.14.
A dual construction would refer to a coalgebra structure, ∆ : A → A ⊗ A, f f ⊗ ⊗ f , and a comodule structure on forms, ρ : Ω → A ⊗ Ω , f dx ∧ dx ... ∧ dx n f ⊗ dx ∧ dx ... ∧ dx n + 1 ⊗ f dx ∧ dx ... ∧ dx n .In this case A ⊕ Ω becomes a coalgebra. Remark 5.15.
The unification of Theorem 5.13 and the construction from Remark5.14 can be realized at the level of Yang-Baxter systems.Take, for example W ( f ⊗ g ) = 1 ⊗ f g and X ( f ⊗ v ) = 1 ⊗ f v to obtain a semiYang-Baxter systems. Conclusions
The Yang–Baxter systems were needed for unifying two constructions.Vr˘anceanu - Vergne basis were topics in a talk at DGA 13 (see also [36]).
Acknowledgment
We would like to thank Prof. Lazlo Stacho and Prof. Nicolae Anghel for their helpand suggestions. Also, we thank the Simion Stoilow Institute of Mathematics of theRomanian Academy and the Petroleum Gas University from Ploiesti.
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