Generalized virtual braid groups, quasi-shuffle product and quantum groups
aa r X i v : . [ m a t h . QA ] N ov GENERALIZED VIRTUAL BRAID GROUPS, QUASI-SHUFFLEPRODUCT AND QUANTUM GROUPS
XIN FANG
Abstract.
We introduce in this paper the generalized virtual braid group on n strands GVB n , generalizing simultaneously the braid groups and their virtualversions. A Mastumoto-Tits type section lifting shuffles in a symmetric group S n to the monoid associated to GVB n is constructed, which is then applied tocharacterize the quantum quasi-shuffle product. A family of representations of GVB n is constructed using quantum groups. Introduction
Quasi-shuffle product appeared firstly in the work of Newman and Radford [14] inaim of constructing the cofree irreducible Hopf algebra on an algebra. This structurewas rediscovered by Hoffman [5] to characterize the product of multiple zeta val-ues (MZVs) which simultaneously encodes the product of quasi-symmetric functions,Rota-Baxter algebras ([2], [4]) and commutative TriDendriform algebras [13].The quasi-shuffle product is quantized by Jian, Rosso and Zhang [7] where a supple-mentary braid structure on the algebra appearing in the framework of Newman andRadford is considered. Recently, in a joint work with Rosso [3], the whole quantumgroup is realized using a properly chosen quantum quasi-shuffle structure.These (quantum) quasi-shuffle products are initially defined either in a functorialway using the universal property or by inductive formulas. Their combinatorialnatures are studied by Guo and Keigher [4] by introducing the mixable shuffles,which is then generalized to the quantum case by Jian [6].One of the goals of this paper is to reveal symmetries behind the quantum quasi-shuffle structure by giving a representation-theoretical point of view on these prod-ucts, which is more natural and computable than mixable shuffles. To archieve thisgoal, we need to introduce the generalized virtual braid groups, generalizing simul-taneously the braid group and its virtual version [9]. After that we construct aMatsumoto-Tits type section (an analogue of the lift of permutations to the braidgroup) lifting ( p, q ) -shuffles from the symmetric group to the generalized virtual braidgroup which encodes the quantum quasi-shuffle product. Since the quasi-shuffle prod-uct is a particular case of its quantization, it is characterized by lifting shuffles to thevirtual braid group. A main ingredient in constructing these generalized Matsumoto-Tits sections is the bubble decomposition introduced in Section 3.Moreover, a family of representations of the generalized virtual braid groups is con-structed using quantum groups and it is hoped to establish some quantum invariantsfor virtual links. Some other perspectives are discussed in the last section.2. Generalized virtual braid groups
Let K be a field of characteristic . All vector spaces, algebras and tensor productsare over K if they are not specified otherwise. Definition 1.
The generalized virtual braid group on n strands GVB n is the groupgenerated by σ i , ξ i for ≤ i ≤ n − which satisfy the following relations: (1). for | i − j | > , σ i σ j = σ j σ i , σ i ξ j = ξ j σ i , ξ i ξ j = ξ j ξ i ; (2). for ≤ i ≤ n − , σ i σ i +1 σ i = σ i +1 σ i σ i +1 , ξ i ξ i +1 ξ i = ξ i +1 ξ i ξ i +1 , ξ i σ i +1 σ i = σ i +1 σ i ξ i +1 , ξ i +1 σ i σ i +1 = σ i σ i +1 ξ i . The group
GVB n can be looked as a double braided group where the two braidingsare compatible.Let S n , B n and VB n denote the symmetric group, the braid group and the virtualbraid group [9] with n letters or on n strands, respectively. The monoids associatedto GVB n , VB n and B n will be denoted by GVB + n , VB + n and B + n .The notion of the generalized virtual braid group enlarges that of the braided groupand its virtual version. Relations between them can be explained in the followingcommutative diagram: GVB n α / / γ (cid:15) (cid:15) B nβ (cid:15) (cid:15) VB n δ / / S n , where α (resp. β ; γ ; δ ) is the quotient by the normal sub-group generated by ξ i (resp. σ i ; σ i ; ξ i ) for ≤ i ≤ n − .3. Generalized Matsumoto-Tits section Bubble decomposition.
The name for this decomposition of permutations insymmetric groups arises from the bubble sort (see [10], Section 5.2.2) which is ineffi-cient in practical use but has interesting theoretical applications.The symmetric group S n with generators s i = ( i, i +1) acts on n sites by permutingtheir positions.We fix an integer n ≥ and p, q ∈ N such that p + q = n . A permutation σ ∈ S n is called a ( p, q ) -shuffle if σ − (1) < · · · < σ − ( p ) and σ − ( p + 1) < · · · < σ − ( p + q ) ;the set of all ( p, q ) -shuffles in S n will be denoted by S p,q .We fix p, q as above. There exists a bijection S n ∼ −→ S p,q · ( S p × S q ) where S p actson the first p positions and S q acts on the last q positions. Iterating this bijectionresults the following bubble decomposition of symmetric groups (a set theoreticalbijection): S n ∼ −→ S n − , × S n − , × · · · × S , × S , where S k, is embedded into S n at the first k + 1 positions.For σ ∈ S n , we let ( σ ( n − , σ ( n − , · · · , σ (1) ) denote its bubble decomposition ac-cording to the above bijection where if σ ( k ) = e , we write σ ( k ) = s t k ( σ ) s t k ( σ )+1 · · · s k ∈ S k, for ≤ t k ( σ ) ≤ k (if σ ( k ) = e , we demand t k ( σ ) = 0 ). Remark 1.
It is easy to show that l ( σ ) = l ( σ (1) ) + · · · + l ( σ ( n − ) , hence the bubbledecomposition gives a reduced expression of σ ∈ S n . We establish several properties of t k ( σ ) when σ is a ( p, q ) -shuffle, which will beuseful in the proof of Theorem 1. Lemma 1.
Let σ ∈ S p,q be a ( p, q ) -shuffle. Then for any ≤ k < p , t k ( σ ) = 0 . ENERALIZED VIRTUAL BRAID GROUPS 3
Proof.
Let n = p + q . We prove by induction on n that for any p , any σ ∈ S p,q and ≤ k < p , t k ( σ ) = 0 .Start by the case n = 3 , the only non-trivial case is p = 2 and q = 1 . In this casewe need to show t ( σ ) = 0 for any σ ∈ S , . This is clear since S , = { e, s , s s } .For general n , we consider the following well-known decomposition of S p,q :(3.1) S p,q = S Rp − ,q ⊔ S Rp,q − s · · · s p where(1) S Rp − ,q and S Rp,q − are images of S p − ,q and S p,q − in S p,q under the inclusion s k s k +1 ;(2) the set S Rp − ,q s · · · s p is defined by { ss · · · s p | s ∈ S Rp − ,q } .Taking σ ∈ S p,q , there are two cases:(1) σ ∈ S Rp − ,q : by induction, for any ≤ k < p , t k ( σ ) = 0 . It remains to showthat t ( σ ) = 0 : this is a consequence of σ ∈ S Rp − ,q .(2) σ ∈ S Rp,q − s · · · s p : we write σ = ss · · · s p for some s ∈ S Rp,q − . By induction,for any ≤ k < p + 1 , t k ( s ) = 0 , therefore for any ≤ k < p , t k ( ss · · · s p ) = t k ( s ) = 0 (in fact, t p ( σ ) = 1 ). It is clear that t ( σ ) = 0 since s ∈ S Rp,q − actson the last p + q − positions. (cid:3) Therefore the bubble decomposition of σ ∈ S p,q admits the form σ = σ ( n − · · · σ ( p ) .The proof of this lemma gives an effective algorithm to compute the bubble de-composition of ( p, q ) -shuffles. Algorithm.
We use the decomposition (3.1) in the proof of Lemma 1 to give arecursive algorithm to compute the bubble decomposition. Suppose that the bubbledecompositions of elements in S p − ,q and S p,q − are known. For a fixed σ ∈ S p,q ,there are two cases:(1) σ ∈ S Rp − ,q : according to the lemma, we may suppose that σ = σ ( n − · · · σ ( p ) is the bubble decomposition of σ in S Rp − ,q . This is also the bubble decompo-sition of σ in S p,q .(2) σ ∈ S Rp,q − s · · · s p : we write σ = ss · · · s p , by Lemma 1 again, we supposethat the bubble decomposition of s in S Rp,q − reads s = s ( n − · · · s ( p +1) , we de-note s ( p ) = s · · · s p , then σ = s ( n − · · · s ( p +1) s ( p ) is the bubble decompositionof σ .This algorithm allows us to prove the following lemma. Lemma 2.
Let σ ∈ S p,q be a ( p, q ) -shuffle and p < s ≤ p + q − . Then t s ( σ ) > s − p .Proof. We apply induction on p + q = n . The case n = 2 is clear.For a general n and σ ∈ S p,q , we need to tackle the following two cases:(1) σ ∈ S Rp − ,q : by induction, for any p − < s ≤ p + q − , t s +1 ( σ ) > s − p + 1 ,therefore t s ( σ ) > s − p for any p < s ≤ p + q − .(2) σ = σ s · · · s p ∈ S Rp,q − s · · · s p : by induction, for any p + 1 < s ≤ p + q − , t s ( σ ) > s − p . According to the second case of the above algorithm, t s ( σ ) >s − p for any p < s ≤ p + q − . (cid:3) The following evident corollary is useful.
XIN FANG
Corollary 1.
Let σ ∈ S p,q be a ( p, q ) -shuffle. If t p ( σ ) = 1 , then for any p ≤ s ≤ p + q − , t s ( σ ) = 1 . Generalized Matsumoto-Tits section.
The numbers n, p, q are fixed as in thelast subsection.We define the map Q p,q : S p,q → K [GVB + n ] by σ k =1 Y k = n,σ ( k ) = e ( σ t k + (1 − δ t k +1 ,t k +1 ) ξ t k ) σ t k +1 · · · σ k , where t k = t k ( σ ) and the product is executed in a descending way. These maps arecalled generalized Matsumoto-Tits section. Example 1.
We take n = 4 and p = q = 2 . The set of (2 , -shuffles S , is givenby { e, s , s s , s s , s s s , s s s s } . According to the bubble decomposition, e ( e, e, e ) , s ( e, s , e ) , s s ( s , s , e ) , s s ( e, s s , e ) ,s s s ( s , s s , e ) , s s s s ( s s , s s , e ) . Their images under the map Q , are: Q , ( e ) = e, Q , ( s ) = σ + ξ , Q , ( s s ) = ( σ + ξ ) σ , Q , ( s s ) = ( σ + ξ ) σ ,Q , ( s s s ) = ( σ + ξ )( σ + ξ ) σ , Q , ( s s s s ) = ( σ + ξ ) σ σ σ . Let e γ : K [GVB + n ] → K [VB + n ] (resp. e α : K [GVB + n ] → K [ B + n ] ) be the quotient bythe ideal generated by σ i − (resp. ξ i ) for ≤ i ≤ n − . Their compositions with Q p,q give two maps V p,q = e γ ◦ Q p,q : S p,q → K [VB + n ] , T p,q = e α ◦ Q p,q : S p,q → K [ B + n ] , where T p,q coincides with the Matsumoto-Tits section S n → B + n (Theorem 4.2 in[8]). In the above example, T , ( s s s ) = σ σ σ .These constructions are summarized in the following commutative diagram: K [ B + n ] K [GVB + n ] e α o o e γ (cid:15) (cid:15) S p,q Q p,q rrrrrrrrrr T p,q O O V p,q / / K [VB + n ] . Application: quantum quasi-shuffle product
We will characterize in this section the quantum quasi-shuffle product using thegeneralized Matsumoto-Tits section.1.
Braided algebra and quantum quasi-shuffle product.
The construction ofquantum quasi-shuffle algebra can be viewed as a functor from the category of braidedalgebras to the category of braided Hopf algebras.
Definition 2.
A braided algebra is a triple ( V, m, σ ) where V is a vector space, m : V ⊗ V → V is an associative operation, σ : V ⊗ V → V ⊗ V is a linear map suchthat on V ⊗ V ⊗ V , the following relations are verified: ( σ ⊗ id)(id ⊗ σ )( σ ⊗ id) = (id ⊗ σ )( σ ⊗ id)(id ⊗ σ ) , ( m ⊗ id)(id ⊗ σ )( σ ⊗ id) = σ ( m ⊗ id) , (id ⊗ m )( σ ⊗ id)(id ⊗ σ ) = σ (id ⊗ m ) . ENERALIZED VIRTUAL BRAID GROUPS 5
Remark 2.
In the definition, σ is not required to be invertible and the algebra ( V, m ) is not required to be unitary. We fix a braided algebra ( V, m, σ ) and let T ( V ) = L n ≥ V ⊗ n to be the tensorvector space built on V . Each V ⊗ n admits a B n -module structure through σ i id ⊗ ( i − ⊗ σ ⊗ id ⊗ ( n − i − . We denote β p, = T p, ( s · · · s p − s p ) ∈ B + p +1 . Definition 3 ([7]) . The quantum quasi-shuffle product is a family of operations ∗ p,q : V ⊗ p ⊗ V ⊗ q → T ( V ) defined by induction as follows: for v , · · · , v p + q ∈ V , ( v ⊗ · · · ⊗ v p ) ∗ p,q ( v p +1 ⊗ · · · ⊗ v p + q )= v ⊗ (( v ⊗ · · · ⊗ v p ) ∗ p − ,q ( v p +1 ⊗ · · · ⊗ v p + q )) ++(id ⊗∗ p,q − )( β p, ⊗ id ⊗ ( q − )( v ⊗ · · · ⊗ v p + q ) ++( m ⊗ ∗ p − ,q − )(id ⊗ β p − , ⊗ id ⊗ ( q − )( v ⊗ · · · ⊗ v p + q ); when p = 0 ou q = 0 , the operation is given by the scalar multiplication. It is proved in [7] that the tensor space T ( V ) , endowed with this family of oper-ations, is an associative algebra. It is called the quantum quasi-shuffle algebra andwill be denoted by T σ,m ( V ) . Example 2. (1)
When p = q = 1 , v ∗ , w = (id + σ + m )( v ⊗ w ) ∈ V ⊕ V ⊗ V . (2) When p = q = 2 , the product ( v ⊗ v ) ∗ , ( v ⊗ v ) is given by the action of id + σ + σ σ + m σ + σ σ + σ σ σ + m σ σ + σ σ σ σ + m σ σ σ + m σ + m σ σ + m m σ on v ⊗ v ⊗ v ⊗ v where σ i and m i signify that σ and m act on the positions i and i + 1 . When m = 0 , the quantum quasi-shuffle product reduces to the quantum shuffleproduct ([16], [17]). When σ = id V ⊗ V , the quantum quasi-shuffle product gives thequasi-shuffle product ([14]), if moreover the multiplication m is commutative, thisproduct is rediscovered by Hoffman in [5] to characterize the product of multiplezeta values (MZVs).The objective of this section is to realize these three products on T ( V ) in a uniformframework.2. A representation-theoretical point of view on braided algebras.
In [11]Lebed realized the associativity of an algebra as an exotic (pre-)braiding, which willbe recalled in the following proposition. This construction allows us to reveal thenature of a braided algebra in the spirit of operads.Let V be a vector space with a specified element ∈ V , m : V ⊗ V → V and ν : K → V be two linear maps where ν sends to ∈ V . We define a linear map σ m : V ⊗ V → V ⊗ V by v ⊗ w ⊗ m ( v ⊗ w ) . Proposition 1 ([11]) . Suppose that for any v ∈ V , m ( v ⊗ ) = v . Then m isassociative if and only if B +3 → End( V ⊗ ) , σ σ m ⊗ id , σ id ⊗ σ m defines arepresentation of B +3 . When a supplementary braiding structure is under consideration, this propositioncan be generalized:
XIN FANG
Proposition 2.
We suppose that for any v ∈ V , m ( v ⊗ ) = v , m ( ⊗ v ) = v , σ ( ⊗ v ) = v ⊗ and σ ( v ⊗ ) = ⊗ v . Then ( V, σ, m ) is a braided algebra if andonly if GVB +3 → End( V ⊗ ) , σ σ ⊗ id , σ id ⊗ σ , ξ σ m ⊗ id , ξ id ⊗ σ m defines a representation of GVB +3 .Proof. It suffices to show that relation in Definition 1 (2) are verified. The secondrelation therein holds by Proposition 1 and the other relations are exactly those inthe definition of a braided algebra. (cid:3) Quantum quasi-shuffle product revisited.
Let ( V, σ, m ) be a braided algebrawith ν : K → V sending to ∈ V which satisfies m ( v ⊗ ) = v, m ( ⊗ v ) = v, σ ( ⊗ v ) = v ⊗ , σ ( v ⊗ ) = ⊗ v. Remark 3.
The requirement of the existence of this specified element ∈ V causes notrouble: given a braided algebra ( V, σ, m ) , enlarging V by e V = K ⊕ V and naturallyextending σ and m solves this problem. According to Proposition 2, for any n ≥ , V ⊗ n admits a representation of GVB + n by σ i id ⊗ ( n − ⊗ σ ⊗ id ⊗ ( n − i − , ξ i id ⊗ ( n − ⊗ σ m ⊗ id ⊗ ( n − i − . Let b V be the linear complement of K in V . Definition 4. (1)
A pure tensor v ⊗ · · · ⊗ v k ∈ T ( V ) is called deletable if forany i = 1 , · · · , k , either v i = or v i ∈ b V . (2) The deleting operator D : T ( V ) → T ( V ) is defined as the linear map deletingall in a deletable pure tensor. For example, D ( ⊗ v ⊗ w ⊗ ⊗ ⊗ u ⊗ ) = v ⊗ w ⊗ u where u, v, w are elementsin b V . Theorem 1.
Let ∗ denote the quantum quasi-shuffle product on T ( V ) . Then for any v , · · · , v p + q ∈ b V , ( v ⊗ · · · ⊗ v p ) ∗ p,q ( v p +1 ⊗ · · · ⊗ v p + q ) = D ◦ X s ∈ S p,q Q p,q ( s )( v ⊗ · · · ⊗ v p + q ) . This theorem will be proved in the next subsection.
Example 3.
The quantum quasi-shuffle product ( v ⊗ v ) ∗ , ( v ⊗ v ) is computedin Example 2, it coincides with the lifting of (2 , -shuffles through the generalizedMatsumoto-Tits section in Example 1. Remark 4. In [4] and [6] , combinatorial formulas on the quasi-shuffle and quantumquasi-shuffle products are proved with the help of mixable shuffles. The combinatorialnature of Theorem 1 differs from those in the papers cited above. Our constructionon the lifting of shuffles relies on the bubble decomposition, which is more naturaland efficiently computable.We explain in the following example the difference between our lifting and themixable shuffles. Consider the (2 , -shuffles: in the context of mixable shuffles, theelement m σ σ corresponds to the (2 , -shuffle s s ; in our framework, as shown inExample 1, it arises from the (2 , -shuffle s s s . As the quasi-shuffle product is a particular case ( σ = id ) of the quantum quasi-shuffle product, composing with the morphism of monoid e γ gives: ENERALIZED VIRTUAL BRAID GROUPS 7
Corollary 2.
Let V be an associative algebra and • be the quasi-shuffle product on T ( V ) ( [14] , [5] ). Then for v , · · · , v p + q ∈ V different from , ( v ⊗ · · · ⊗ v p ) • p,q ( v p +1 ⊗ · · · ⊗ v p + q ) = D ◦ X s ∈ S p,q V p,q ( s )( v ⊗ · · · ⊗ v p + q ) . These formulas generalize those for quantum shuffle product ♦ ([16], [17]): for abraided vector space ( V, σ ) , ( v ⊗ · · · ⊗ v p ) ♦ p,q ( v p +1 ⊗ · · · ⊗ v p + q ) = X s ∈ S p,q T p,q ( s )( v ⊗ · · · ⊗ v p + q ) . Proof of Theorem 1.
We prove it by induction on n = p + q . The case p + q = 1 is clear.By the inductive definition of the quantum quasi-shuffle product in Definition 3and the induction hypothesis, it suffices to show that X s ∈ S p,q Q p,q ( s ) = X s ∈ S Rp − ,q Q p − ,q ( s )+ X s ∈ S Rp,q − Q p,q − ( s ) σ · · · σ p + X s ∈ S Rp − ,q − Q p − ,q − ( s ) ξ σ · · · σ p , where the notations S Rp,q − and S Rp − ,q are explained in the proof of Lemma 1. Theset S Rp − ,q − is the image of S p − ,q − in S p,q under the inclusion s i s i +2 for ≤ i ≤ p + q − .We divide S p,q into three disjoint subsets S p,q = S ⊔ S ⊔ S where S = { σ ∈ S p,q | t p ( σ ) = 1 } , S = { σ ∈ S p,q | t p ( σ ) = 1 , t p +1 ( σ ) = 2 } ,S = { σ ∈ S p,q | t p ( σ ) = 1 , t p +1 ( σ ) = 2 } . We will in fact prove the following three identities, then taking the sum impliesthe formula desired:(4.1) X σ ∈ S Q p,q ( σ ) = X σ ∈ S Rp − ,q Q p − ,q ( σ ) , (4.2) X σ ∈ S Q p,q ( σ ) = X σ ∈ S Rp,q − \ S Rp − ,q − Q p,q − ( σ ( n − · · · σ ( p +1) ) σ · · · σ p , (4.3) X σ ∈ S Q p,q ( σ ) = X σ ∈ S Rp − ,q − Q p − ,q − ( σ ( n − · · · σ ( p +1) )( σ + ξ ) σ · · · σ p . We start by giving some combinatorial explanations of these three sets.(1) We show that S = { σ ∈ S p,q | σ (1) = 1 } . That is to say, for any ( p, q ) -shuffle σ , σ (1) = 1 if and only if t p ( σ ) = 1 . According to Lemma 1 and Corollary1, if t p ( σ ) = 1 , then s does not appear in a reduced expression of σ , fromwhich σ (1) = 1 . On the other hand, if σ (1) = 1 , then t p ( σ ) = 1 by Lemma 2.(2) We prove that S = ( S Rp,q − \ S Rp − ,q − ) s · · · s p .Let σ ∈ S , which means that σ ∈ S p,q , t p ( σ ) = 1 and t p +1 ( σ ) = 2 .The condition t p ( σ ) = 1 implies that σ ( p ) = s · · · s p . From the bubbledecomposition and Lemma 1, σ = σ ( n ) · · · σ ( p +1) s · · · s p . We show that σ ′ = σ ( n ) · · · σ ( p +1) ∈ S Rp,q − \ S Rp − ,q − : to show that it is not in S Rp − ,q − , XIN FANG whose all elements in S p,q − preserving the position , we consider t p +1 ( σ ) :if t p +1 ( σ ) = 2 , then σ ( p +1) = s · · · s p +1 will not preserve the position ; byLemma 2, so does σ ′ .We take σ ∈ S p,q with bubble decomposition σ = σ ( n − · · · σ ( p +1) s · · · s p where σ ′ = σ ( n − · · · σ ( p +1) ∈ S Rp,q − \ S Rp − ,q − . Since σ ′ / ∈ S Rp − ,q − , t p ( σ ) = 1 .Moreover, σ ′ does not preserve the position , then Lemma 2 forces σ ( p +1) = s · · · s p +1 , from which t p +1 ( σ ) = 2 .(3) Finally we claim that S = S Rp − ,q − s · · · s p . As S Rp,q − s · · · s p = { σ ∈ S p,q | t p ( σ ) = 1 } , the desired identity is the complement of the one proved in(2).By definition, for σ ∈ S , Q p,q ( σ ) = Q p − ,q ( σ ) and therefore (4.1) holds. Let σ ∈ S .It admits the decomposition σ = σ ( n − · · · σ ( p +1) s · · · s p , since t p +1 ( σ ) = 2 , in thedefinition of the generalized Matsumoto-Tits section, terms including the Kronecker δ vanish, it therefore gives Q p,q ( σ ) = Q p,q − ( σ ( n − · · · σ ( p +1) ) σ · · · σ p . This proves (4.2). Finally, for σ ∈ S , the same argument as above shows (4.3). Theproof of Theorem 1 terminates.5. Representations of generalized virtual braid groups and quantumgroups
We want to construct representations of the generalized virtual braid group viaquantum groups: it requires us to investigate the other structure inside of a quantumgroup encoding the virtual crossings.Let V be a vector space. We let P : V ⊗ V → V ⊗ V denote the usual flip definedby P ( v ⊗ w ) = w ⊗ v for v, w ∈ V .1. Quasi-triangular Hopf algebras with a twist.
This structure we desired isfirstly studied by Drinfeld in quasi-Hopf algebras and then by Reshetikhin [15] in theearly stage of quantum groups.
Definition 5 ([15]) . Let H be a Hopf algebra and R, F ∈ H ⊗ H be two invertibleelements. The triple ( H, R, F ) is called a quasi-triangular Hopf algebra with a twist(QTHAT) if (1) ( H, R ) if a quasi-triangular Hopf algebra (QTHA) ( [1] ); (2) (∆ ⊗ id)( F ) = F F , (id ⊗ ∆)( F ) = F F , F F F = F F F where F = F ⊗ , F = 1 ⊗ F et F = P a i ⊗ ⊗ b i if F = P a i ⊗ b i . Such an F is called a twist.If moreover F satisfies F F = 1 ⊗ where F = P b i ⊗ a i , the triple is calledinvolutive. For example, any quasi-triangular Hopf algebra has a twist F = 1 ⊗ .In [15], only involutive triples are considered. It is interesting to find non-involutiveQTHATs.The following lemma is well-known, whose proof is a standard exercise. Lemma 3.
Let ( H, R, F ) be a QTHAT. Then R F F = F F R , F F R = R F F . ENERALIZED VIRTUAL BRAID GROUPS 9
The QTHATs allow us to construct tensor representations of the generalized virtualbraid groups.
Proposition 3.
Let ( V, ρ ) be a representation of a QTHAT ( H, R, F ) . We denote ˇ R = P ◦ ( ρ ⊗ ρ )( R ) and ˇ F = P ◦ ( ρ ⊗ ρ )( F ) . Then µ : GVB n → GL( V ⊗ n ) ,σ i id ⊗ ( i − ⊗ ˇ R ⊗ id ⊗ ( n − i − , ξ i id ⊗ ( i − ⊗ ˇ F ⊗ id ⊗ ( n − i − defines a representation of the group GVB n . Moreover, if the triple ( H, R, F ) isinvolutive, µ factorizes through VB n and gives a representation of the virtual braidgroup VB n .Proof. This proposition is a consequence of the definition of a QTHAT and Lemma3. (cid:3) Example: quantum groups.
Let g be a finite dimensional semi-simple Liealgebra of rank l over C and U ~ ( g ) be the ~ -adic version of the quantum groupassociated to g ([1]) with generators E i , F i and H i for i = 1 , · · · , l .The Hopf algebra U ~ ( g ) is quasi-triangular with the universal R-matrix R ([1]).Reshetikhin constructed in [15] a non-trivial element F ∈ U ~ ( g ) ⊗ U ~ ( g ) : F = exp X i Let W be a finite dimensional representation of the quantum group U ~ ( g ) . Then for any n ≥ , W ⊗ n admits a GVB n -module structure which is factor-izable through VB n . Remark 5. The categorical interpretation of the virtual braids [12] can be generalizedto the case of this paper to obtain the notion of the double braided categories. Thecategory of finite dimensional representations of the quantum group U ~ ( g ) admitssuch a structure. Problem 1. Theorem 2 serves as the starting point of the quantum invariant theoryof the virtual links. That is to say, we want to know whether the construction ofTuraev [18] can be modified to fit for the virtual links. Problem 2. The virtual Hecke algebra VH n ( q ) is a quotient of the generalized virtualbraid group GVB n by relations: ξ i = 1 , σ i = ( q − σ i + q (equivalently, it is aquotient of the virtual braid group VB n by the relation σ i = ( q − σ i + q ). Problem1 is equivalent to the study of the existence of a virtual Markov trace on VH n ( q ) .Moreover, it would be interesting to find the object having Schur-Weyl duality with VH n ( q ) , it is expected to be some object interpolating the classical gl n and quantum sl n . Problem 3. As the universal R-matrix admits the uniqueness property. It is naturalto ask for a classification of the invertible elements F ∈ U ~ ( g ) b ⊗ U ~ ( g ) satisfying thepoint (2) in Definition 5 and involutive. Acknowledgements I would like to thank Run-Qiang Jian for stimulating discussions. This work ispartially supported by the National Natural Science Foundation of China (GrantNo. 11201067). References [1] V. Drinfel’d, Quantum groups , Proceedings of the International Congress of Mathematicians,Vol. 1, 2 (Berkeley, Calif., 1986), 798-820, Amer. Math. Soc., Providence, RI, 1987.[2] K. Ebrahimi-Fard, L. Guo, Mixable Shuffles, Quasi-shuffles and Hopf Algebras, J. AlgebraicCombinatorics, , no 1, (2006), 83-101.[3] X. Fang, M. Rosso, Multi-brace cotensor Hopf algebras and quantum groups, preprint,arXiv:1210.3096.[4] L. Guo, W. Keigher, Baxter algebras and shuffle products, Adv. Math. , (2000), 117-149.[5] M.E. Hoffman, Quasi-shuffle products . J. Algebraic Combin. (2000), no. 1, 49-68.[6] R. Jian, Quantum quasi-shuffle algebras II. Explicit formulas, dualization, and representations, preprint, arXiv:1302.5888.[7] R. Jian, M. Rosso, J. Zhang, Quantum quasi-shuffle algebras, Lett. Math. Phys. (2010),1-16.[8] Ch. Kassel, V. Turaev, Braid groups, Graduate Texts in Mathematics , Springer, New York(2008).[9] Louis H. Kauffman, S. Lambropoulou, Virtual braids , Fund. Math. (2004), 159-186.[10] D. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching , Third Edi-tion. Addison-Wesley, 1997.[11] V. Lebed, Homologies of Algebraic Structures via Braidings and Quantum Shuffles, Journal ofAlgebra, (2013), 152-192.[12] V. Lebed, Categorical Aspects of Virtuality and Self-Distributivity , to appear in Journal of KnotTheory and Its Ramifications.[13] J.-L. Loday, On the algebra of quasi-shuffles , Manuscripta Mathematica (2007), no. 1,79-93.[14] K. Newman, D. Radford, The cofree irreducible Hopf algebra on an algebra , Amer. J. Math. (1979), no. 5, 1025-1045.[15] N. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras ,Letters in Mathematical Physics, 1990, , Issue 4, 331-335.[16] M. Rosso, Groupes quantiques et algèbres de battage quantiques , C. R. Acad. Sci. Paris Sér. IMath. (1995), no. 2, 145-148.[17] M. Rosso, Quantum groups and quantum shuffles , Invent. Math. , (1998), 399-416.[18] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. (1988), no.3, 527-553. Institut des Hautes Études Scientifiques, 35, route de Chartes, Bures-sur-Yvette91440, France. E-mail address ::