Set theoretic Yang-Baxter equation, braces and Drinfeld twists
aa r X i v : . [ m a t h - ph ] F e b SET THEORETIC YANG-BAXTER EQUATION, BRACES ANDDRINFELD TWISTS
ANASTASIA DOIKOU
Abstract.
We consider involutive, non-degenerate, finite set theoretic solu-tions of the Yang-Baxter equation. Such solutions can be always obtained us-ing certain algebraic structures that generalize nil potent rings called braces.Our main aim here is to express such solutions in terms of admissible
Drinfeldtwists substantially extending recent preliminary results. We first identify thegeneric form of the twists associated to set theoretic solutions and we showthat these twists are admissible, i.e. they satisfy a certain co-cycle condition.These findings are also valid for Baxterized solutions of the Yang-Baxterequation constructed from the set theoretical ones. Introduction
The idea of set theoretic solutions of the Yang-Baxter equation (YBE) was firstsuggested by Drinfield [9] and since then there has been a considerable researchactivity on this topic from the algebraic point of view [19, 12, 13], but also inthe context of classical integrable systems [1, 30, 25]. More recently, an algebraicstructure that generalizes nil potent rings, called a brace, was introduced [26, 27],to describe all finite, involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation. Rump showed [26, 27] that every brace yields a solution to theYang-Baxter equation, and every non-degenerate, involutive set-theoretic solutionof the Yang-Baxter equation can be obtained from a brace. Subsequently skew-braces were developed in [18] to describe non-involutive solutions. This emergingresearch area has been particularly fruitful and numerous relevant studies havebeen produced over the past few years (see for instance [2, 4, 5], [15]-[17], [18, 20,28, 29]).In [7, 8] key links between set theoretical solutions coming from braces andquantum integrable systems and the associated quantum algebras were uncovered.More precisely:
Date : March 1, 2021.
Key words and phrases.
Yang-Baxter equation, braces, quantum groups, Drinfeld twists. (1) Quantum groups associated to Baxterized solutions of the Yang-Baxterequations coming from braces were derived via the FRT construction [14].(2) Novel classes of quantum discrete integrable systems with periodic andopen boundary conditions were produced.(3) Symmetries of the periodic and open transfer matrices of the novel inte-grable systems were identified.(4) Preliminary findings on Drinfeld twists for set theoretic solutions werepresented.Note that in [12] Etingof, Schedler and Soloviev constructed quantum groups as-sociated to set-theoretic solutions, however in [7] a different construction is usedcoming from parameter dependent solutions of the Yang-Baxter equation and thusthe corresponding quantum groups differ from those in [12].Our main focus here is the study of involutive set theoretic solutions comingfrom braces, based on the findings presented in [7, 8]. More precisely, our aim is toidentify the explicit form of so called admissible Drinfeld twists for brace solutionsof the YBE. We note that in Drinfeld’s original works on quantum algebras [10, 11]it was required that the universal R -matrix, solution of the Yang-Baxter equationhas a semi-classical limit, i.e. it can be expressed as formal series expansion, R = 1 + P ∞ n =1 h n R ( n ) , where h is some “deformation” parameter. In [11] Drinfeldintroduced the notion of the twist F , which links distinct solutions of the Yang-Baxter equation, and consequently distinct quantum algebras, provided that thetwist is admissible . Such twists have also semi-classical limits [11], i.e. they canbe expressed as formal series expansions in powers of the deformation parameter,with the leading term being the identity.In the analysis of [7, 8] Baxterized R -matrices coming from set theoretic solu-tions of the Yang-Baxter equation were identified, being of the form R ( λ ) = r + λ P ,where r is the set theoretic solution of the Yang-Baxter equation and P is the per-mutation operator. Interestingly the r -matrix does not contain any free parame-ter (deformation parameter), and consequently the R -matrix has no semi-classicalanalogue. This is the pivotal difference between our study here and Drinfeld’sanalysis [11]. A similar observation can be made about the associated Drinfeldtwist, which is explicitly identified in the present study. To be more precise themain results of this investigation, presented in section 3, are:(1) The identification of the explicit form of “local” and “global” Drinfeldtwists.(2) The proof of the admissibility of Drinfeld’s twists associated to involutive,non-degenerate, set theoretic solutions of the Yang-Baxter equation. ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 3
The outline of the paper.
We describe below what is achieved in each of thesubsequent sections. • In section 2 we present necessary preliminary notions regarding the settheoretic solutions of the YBE and the associated quantum groups derivedvia the FRT construction [7]. More specifically:(1) In subsection 2.1 we recall the definition of the set theoretic YBE.We also recall Rump’s theorem, which states that every involutive,non-degenerate, set theoretic solution of the YBE comes from a braceand vice versa.(2) In subsection 2.2 we review the construction of the quantum groupassociated to involutive, set theoretic solutions by means of the FRTformulation. • In section 3 we first recall basic definitions and results on Hopf algebrasand Drinfeld twists [10, 11]. We then move on with the presentation ofour main findings regarding the derivation of admissible Drinfeld twistsassociated to set theoretic solutions of the YBE. Specifically:(1) In subsection 3.1 we review basic background on bialgebras, Hopfand quasi-triangular Hopf algebras. We then introduce the notionof admissible Drinfeld twist and recall fundamental propositions onadmissible twists adjusted for your purposes here.(2) In subsection 3.2 we present the new findings on the Drinfeld twistsfor involutive, non-degenerate, set theoretic solutions. To achieve thiswe rely on the results of [8], where the set theoretic braid r -matrixwas obtained from the permutation operator via a similarity transfor-mation. We use the similarity transformation to derive explicit formsfor the set theoretic twist. We also derive co-product expressions ofthe twist, which enable the derivation of the n -twist. From the de-rived explicit expressions we are able to show the admissibility of thetwist i.e. we show that it satisfies the co-cycle condition. Note thatwe introduce the term quasi-admissible twist given that we only showthe admissibility of the set theoretic twists, without exploring the no-tions of the co-unit and antipode, although some related commentsare presented at the end of subsection 3.3.(3) In subsection 3.3 we focus on a simple, but characteristic example ofset theoretic solution of the YBE known as Lyubashenko’s solution[9]). We first introduce this class of solutions and we show that theycan be expressed as Reshetikhin type twists recalling the results of[8]. We then move on to show that this is an admissible twist and wederive simple expressions for the n -twist. At the end of this subsection ANASTASIA DOIKOU we present some preliminary observations related to the action of theco-unit on the twisted co-products. We employ the Lyubashenkosolution to illustrate the non-trivial action of the co-unit.2.
Preliminaries
We present in this section basic background information regarding set theoreticsolutions of the Yang-Baxter equation and braces as well as a brief review on therecent findings of [7] on the links between set theoretic solutions of the Yang-Baxterequation from braces and quantum algebras.2.1.
The set theoretic Yang-Baxter equation.
Let X = { x , . . . , x n } be a setand ˇ r : X × X → X × X . Denoteˇ r ( x, y ) = (cid:0) σ x ( y ) , τ y ( x ) (cid:1) . We say that r is non-degenerate if σ x and τ y are bijective functions. Also, thesolutions ( X, ˇ r ) is involutive: ˇ r ( σ x ( y ) , τ y ( x )) = ( x, y ), (ˇ r ˇ r ( x, y ) = ( x, y )). Wefocus on non-degenerate, involutive solutions of the set theoretic braid equation:(ˇ r × id X )( id X × ˇ r )(ˇ r × id X ) = ( id X × ˇ r )(ˇ r × id X )( id X × ˇ r ) . Let V be the space of dimension equal to the cardinality of X , and with a slightabuse of notation, let ˇ r also denote the R -matrix associated to the linearisation ofˇ r on V = C X (see [29] for more details), i.e. ˇ r is the N × N matrix:(2.1) ˇ r = X x,y,z,w ∈ X ˇ r ( x, z | y, w ) e x,z ⊗ e y,w , where e x,y is the N ×N matrix: ( e x,y ) z,w = δ x,z δ y,w . Then for the ˇ r -matrix relatedto ( X, ˇ r ): ˇ r ( x, z | y, w ) = δ z,σ x ( y ) δ w,τ y ( x ) . Notice that the matrix ˇ r : V ⊗ V → V ⊗ V satisfies the (constant) Braid equation:(ˇ r ⊗ I V )( I V ⊗ ˇ r )(ˇ r ⊗ I V ) = ( I V ⊗ ˇ r )(ˇ r ⊗ I V )( I V ⊗ ˇ r ) . Notice also that ˇ r = I V ⊗ V the identity matrix, because ˇ r is involutive.For set theoretic solutions it is thus convenient to use the matrix notation:(2.2) ˇ r = X x,y ∈ X e x,σ x ( y ) ⊗ e y,τ y ( x ) . Define also, r = P ˇ r , where P = P x,y ∈ X e x,y ⊗ e y,x is the permutation operator,consequently r = P x,y ∈ X e y,σ x ( y ) ⊗ e x,τ y ( x ) . The Yangian [31] is a special case:ˇ r ( x, z | y, w ) = δ z,y δ w,x .Let us now recall the role of braces in the derivation of set theoretic solutionsof the Yang-Baxter equation. In [26, 27] Rump showed that every solution ( X, ˇ r )can be in a good way embedded in a brace. ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 5
Definition 2.1 (Proposition 4, [27]) . A left brace is an abelian group ( A ; +) together with a multiplication · such that the circle operation a ◦ b = a · b + a + b makes A into a group, and a · ( b + c ) = a · b + a · c . In many papers, an equivalent definition is used [5] . The additive identity of abrace A will be denoted by 0 and the multiplicative identity by 1. In every brace0 = 1. The same notation will be used for skew braces (in every skew brace 0 = 1).Throughout this paper we will use the following result, which is implicit in[26, 27] and explicit in Theorem 4.4 of [5]. Theorem 2.2. (Rump’s theorem, [26, 27, 5] ). It is known that for an involutive,non-degenerate solution of the braid equation there is always an underlying brace ( B, ◦ , +) , such that the maps σ x and τ y come from this brace, and X is a subsetin this brace such that ˇ r ( X, X ) ⊆ ( X, X ) and ˇ r ( x, y ) = ( σ x ( y ) , τ y ( x )) , where σ x ( y ) = x ◦ y − x , τ y ( x ) = t ◦ x − t , where t is the inverse of σ x ( y ) in the circlegroup ( B, ◦ ) . Moreover, we can assume that every element from B belongs to theadditive group ( X, +) generated by elements of X . In addition every solution ofthis type is a non-degenerate, involutive set-theoretic solution of the braid equation. We will call the brace B an underlying brace of the solution ( X, ˇ r ), or a braceassociated to the solution ( X, ˇ r ). We will also say that the solution ( X, ˇ r ) isassociated to brace B . Notice that this is also related to the formula of set-theoretic solutions associated to the braided group (see [12] and [16]).2.2. Yang Baxter equation & quantum groups. In this subsection we brieflyreview the main results reported in [8] on the various links between braces, repre-sentations of the A -type Hecke algebras, and quantum algebras.Recall first the Yang-Baxter equation in the braid form ( δ = λ − λ ):(2.3) ˇ R ( δ ) ˇ R ( λ ) ˇ R ( λ ) = ˇ R ( λ ) ˇ R ( λ ) ˇ R ( δ ) . We focus on brace solutions of the YBE, given by (2.2) and Baxterized solutions:(2.4) ˇ R ( λ ) = λ ˇ r + I , where I = I X ⊗ I X and I X is the identity matrix of dimension equal to thecardinality of the set X . Let also R = P ˇ R , (recall the permutation operator P = P x,y e x,y ⊗ e y,x ), then the following basic properties for R matrices comingfrom braces were shown in [8]: ANASTASIA DOIKOU
Basic Properties.
The brace R -matrix satisfies the following fundamental prop-erties: R ( λ ) R ( − λ ) = ( − λ + 1) I , Unitarity (2.5) R t ( λ ) R t ( − λ − N ) = λ ( − λ − N ) I , Crossing-unitarity (2.6) R t t ( λ ) = R ( λ ) , where t , denotes transposition on the first, second space respectively. The brace solution ˇ r (2.2) is a representation of the A -type Hecke algebra for q = 1 (see also [8]). The quantum algebra.
Our approach on deriving the quantum groups associ-ated to set theoretic solutions [7, 8] is based on the FRT construction [14], which issomehow dual to the Hopf algebraic description [10]. Indeed, the FRT constructioncan be considered as an inverse procedure of the one that uses the quasi-triangularHopf algebras axioms in obtaining a solution of the YBE. The FRT constructionconsiders a solution R : V ⊗ V → V ⊗ V ( V is usually a finite vector space) of theYBE as an input and produces a bialgebra as an output.Given a solution of the Yang-Baxter equation, the quantum algebra is definedvia the fundamental relation [14] (we have multiplied the familiar RTT relationwith the permutation operator):(2.7) ˇ R ( λ − λ ) L ( λ ) L ( λ ) = L ( λ ) L ( λ ) ˇ R ( λ − λ ) . ˇ R ( λ ) ∈ End( V ⊗ V ), L ( λ ) ∈ End( V ) ⊗ A , where A is the quantum algebra definedby (2.7). We focus on solutions associated to braces given by (2.4), (2.2). Thedefining relations of the corresponding quantum algebra were derived in [7]. The quantum algebra associated to the brace R matrix (2.4), (2.2) is defined bygenerators L ( m ) zw , z, w ∈ X , and defining relations L ( n ) z,w L ( m )ˆ z, ˆ w − L ( m ) z,w L ( n )ˆ z, ˆ w = L ( m ) z,σ w ( ˆ w ) L ( n +1)ˆ z,τ ˆ w ( w ) − L ( m +1) z,σ w ( ˆ w ) L ( n )ˆ z,τ ˆ w ( w ) − L ( n +1) σ z (ˆ z ) ,w L ( m ) τ ˆ z ( z ) , ˆ w + L ( n ) σ z (ˆ z, ) w L ( m +1) τ ˆ z ( z ) , ˆ w . (2.8)The proof is based on the fundamental relation (2.7) and the form of the brace R - matrix (for the detailed proof see [8]). Recall also that in the index notationwe define ˇ R = ˇ R ⊗ id A : L ( λ ) = X z,w ∈ X e z,w ⊗ I ⊗ L z,w ( λ ) , L ( λ ) = X z,w ∈ X I ⊗ e z,w ⊗ L z,w ( λ )(2.9) All, finite, non-degenerate, involutive, set theoretic solutions of the YBE are coming frombraces, therefore we will call such solutions brace solutions . ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 7 where L z,w ( λ ) are the generators of the affine algebra A and ˇ R is given in (2.4),(2.2). The quantum algebra is equipped with a co-product ∆ : A A ⊗ A suchthat (id ⊗ ∆) L ( λ ) = L ( λ ) L ( λ ) and (∆ ⊗ id) L ( λ ) = L ( λ ) L ( λ ) , [14, 10].3. Set theoretic solutions as Drinfeld twists
In this section we recall basic definitions and results on Hopf algebras and Drin-feld twists [10, 11], we then move on with the presentation of our main findingsregarding the derivation of admissible Drinfeld twists associated to set theoreticsolutions of the YBE.3.1.
Quasi-triangular Hopf algebras & Drinfeld twists.
Before we presentour brief review on Drinfeld’s twists let us first recall the notion of a quasi-triangular Hopf algebra. The known quantum algebras defined by the RTT relation(2.7) are quasi-triangular Hopf algebras (on a detailed discussion on bialgebras,Hopf algebras, and quasi-triangular Hopf algebras we refer the interested readerfor instance in [10, 11, 6, 24, 23]). We give below some useful definitions regarding,bialgebras, Hopf and quasi-triangular Hopf algebras (see also for instance [6, 24]).
Definition 3.1.
A bialgebra H is a vector space over some field k with linearmaps: (1) m : H ⊗ H → H , multiplication, m ( a ⊗ b ) = ab , which is associative, i.e. ( ab ) c = a ( bc ) ∀ a, b, c ∈ H. (2) η : k → H such that it produces the unit element for H , η (1) = 1 H ( η ( c ) = c · H ). (3) ∆ : H → H ⊗ H , coproduct ∆( a ) = P j a j ⊗ b j ) which is coassociative, i.e. (∆ ⊗ id )∆ = ( id ⊗ ∆)∆ . (4) ǫ : H → k , counit such that ( ǫ ⊗ id )∆( a ) = ( id ⊗ ǫ )∆( a ) = a , ∀ a ∈ H . (5) ∆ , ǫ are algebra homomorphisms and H ⊗ H has the structure of a tensorproduct algebra: ( a ⊗ b )( c ⊗ d ) = ac ⊗ bd , ∀ a, b, c, d ∈ H. In other words, (
H, m, η ) is an associative algebra and ( H, ∆ , ǫ ) is an associa-tive coalgebra. The compatibility of the bialgebra axioms is typically representedby commutative diagrams (see for instance [6, 24] and references therein). Definition 3.2.
A Hopf algebra H is a bialgebra equipped with a bijective linearmap, the “antipode”, s : H → H, such that m (cid:0) ( s ⊗ id )∆( g ) (cid:1) = m (cid:0) ( id ⊗ s )∆( g ) (cid:1) = ǫ ( g ) · H , ∀ g ∈ H. We may now give the definition of a quasi-triangular Hopf algebra, which is ourmain focus in here.
ANASTASIA DOIKOU
Definition 3.3.
Let A be a Hopf algebra over some field k , then A is a quasi-triangular Hopf algebra if there exists an invertible element R ∈ A ⊗ A : (1) R ∆( a ) = ∆ op ( a ) R , ∀ a ∈ A ,where ∆ : A → A ⊗ A is the co-product on A and ∆ op ( a ) = π ◦ ∆( a ) , π : A → A such that π ( a ⊗ b ) = b ⊗ a . (2) ( id ⊗ ∆) R = R R , and (∆ ⊗ id ) R = R R . Also, the following statements hold: • The antipode s : A → A satisfies (id ⊗ s ) R − = R , ( s ⊗ id) R = R − , • The co-unit ǫ : A → k satisfies (id ⊗ ǫ ) R = ( ǫ ⊗ id) R = id. • Due to (1) and (2) of Definition 3.3 the R -matrix satisfies the Yang-Baxterequation R R R = R R R .Proofs of the above statements can be found for instance in [6, 24]. Note thata sufficient condition for the co-associativity of ∆ ( op ) is ( R ⊗
1) (∆ ⊗ id) R =(1 ⊗ R ) (id ⊗ ∆) R , i.e. R should satisfy the co-cycle condition. The co-cyclecondition together with (2) of Definition 3.3 lead to YBE.The index notation holds in this case as follows. Let R = P j a j ⊗ b j ∈ A ⊗ A then R = P j a j ⊗ b j ⊗ R = P j ⊗ a j ⊗ b j etc.If in addition to the above conditions RR ( op ) = 1 A⊗A , where R ( op ) = σ ◦ R ,(or in the index notation R R = 1), then A is a triangular Hopf-algebra Drinfelds twists.
In [10, 11] Drinfeld introduced the notion of twisting (or defor-mation) for quasi-Hopf algebras and quasi-triangular quasi-Hopf algebras. We fo-cus here on quasi-triangular Hopf algebras. Let A , equipped with ( m, η, ǫ, ∆ , s, ǫ ),be a quasi-triangular Hopf algebra over C and F ∈ A ⊗ A be an invertible elementsuch that (id ⊗ ǫ ) F = ( ǫ ⊗ id) F = 1. The twist of A generated by F defines a newquasi-triangular Hopf algebra ˜ A , equipped with ( m, η, ǫ, ˜∆ , ˜ s, ˜ ǫ ) , having the sameelements and product law as A , and such that:(3.1) ˜ R = F op R F − , where we define F op = π ◦ F , and the permutation (flip) map π : a ⊗ b b ⊗ a . Inthe index notation F op = F . Also, u = m (cid:0) (id ⊗ s ) F (cid:1) is an invertible element in A with u − = m (cid:0) ( s ⊗ id) F − (cid:1) , and the twisted maps, ˜∆ and ˜ s are defined as(3.2) ˜∆( a ) = F ∆( a ) F − , ˜ s = us ( a ) u − . Definition 3.4.
A twist F is called admissible if it satisfies the co-cycle condition: (3.3) ( F ⊗
1) (∆ ⊗ id ) F = (1 ⊗ F ) ( id ⊗ ∆) F . In the special case that ˜ R = 1 , then R = ( F ( op ) ) − F and such a twist is called factorizable . ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 9
We now introduce some convenient index notation:(3.4) F , := (∆ ⊗ id) F , F , := (id ⊗ ∆) F . Then according to the co-cycle conditions above we introduce F = F F , = F F . . In general, we define(3.5) F ...n,n +1 := (∆ ( n ) ⊗ id) F , F , ...n +1 := (id ⊗ ∆ ( n ) ) F and consequently we define the n -twist, compatible also with the co-cycle condi-tion:(3.6) F ...n = F ...n − F ..n − ,n = F ..n F , ..n . We introduce a similar notation for tensor representations of the quantum algebra(3.7) T , ..n := (id ⊗ ∆ ( n ) ) R where recall (id ⊗ ∆) R = R R and (∆ ⊗ id) = R R , thus we can explicitlywrite T , ...n = R n . . . R R . The n + 1 co-cycle condition and the explicitexpressions of the n + 1 twist are also useful for our purposes here: F ...n = F F , F , . . . F ...n − ,n = F n − n F n − ,n − n . . . F , ..n − n . (3.8)We state below three Propositions that include some of the main Drinfeld’sresults [10, 11] restricted to R -matrices that satisfy the Yang-Baxter equation.These results are most relevant to our present analysis that follows in the nextsubsection. We also refer the interested reader to [23] and references therein onsimilar proofs using the index notation. Proposition 3.5. (Drinfled 1) Let R be a solution of the Yang-Baxter equationand F be an admissible twist such that, in the index notation, ˜ R n = F n R n F − n , where n ∈ { , . . . N } , then (3.9) F ...n − n n +1 ...N R n = ˜ R n F ...n − nn +1 ...N . Proof.
The proof relies on three basic statements:(1) ˜ R n = F n R n F − n (2) F m − ,m...n n +1 ...N R n = R n F m − ,m...n − n...N , m ≤ n ∈ { , . . . N } and F ...n n +1 ...m,m +1 R n = R n F ...n − n...m,m +1 ,m ≥ n ∈ { , . . . N } (3) the generalized co-cycle condition (3.8) Statement (2) is a natural consequence of (3.5) and the fact that R satisfies theYang-Baxter equation or equivalently conditions (1) and (2) of Definition 3.3.We start with F ...n − n n +1 ...N R n = F ...n − n F ...n − n ,n +1 . . . F ...n − n n +1 ...N − ,N R n = F ...n − n R n F ...n − n,n +1 . . . F ...n − nn +1 ...N − ,N , (3.10)where we have repeatedly used the generalized expressions (3.8) and statement(2). We now focus on F ...n − n R n = F n F n − ,n . . . F , ...n R n = F n R n F n − , n . . . F , ... n = ˜ R n F n F n − , n . . . F , ... n = ˜ R n F ... n . (3.11)Then by means of (3.37) statement (1) and (3.8) expression (3.10) becomes˜ R n F ...n − nn +1 ...N , which concludes our proof. (cid:3) We may now present the next two basic Propositions.
Proposition 3.6. (Drinfeld 2) Let R satisfy the Yang-Baxter equation and ˜ R = F op R F − , where F is an admissible twist, then ˜ R also satisfies the YBE.Proof. We are employing the index notation together with Lemma (3.5). R satis-fies the YBE, we multiply YBE with F form the left and F − from the right:(3.12) F ( R R R = R R R ) F − . We focus on the left side of the equation above, and use Lemma 3.5: F R R R F − = ˜ R F R R F − = ˜ R ˜ R F R F − =˜ R ˜ R ˜ R F F − = ˜ R ˜ R ˜ R . Similarly, for the right hand side of (3.12) we end up with ˜ R ˜ R ˜ R , and thisconcludes our proof. (cid:3) Proposition 3.7. (Drinfeld 3) Let R be a solution of the Yang-Baxter equationand F be an admissible twist: ˜ R = F op R F − . Let also F ...N defined in (3.8), T , ..N := R N . . . R and ˜ T , ...N := ˜ R N . . . ˜ R , then (3.13) F ...N T , ..N F − ..N = ˜ T , ...N ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 11
Proof.
The proof of the this Proposition follows exactly the same logic as the proofof Proposition 3.6. We start with the LHS of (3.13), and use Proposition 3.5: F ...N R N R N − . . . R F − ..N =˜ R N F ...N − N R N − . . . R F − ..N = . . . ˜ R N ˜ R N − . . . ˜ R F ...N − N F − ..N ≡ ˜ T . ...N . (cid:3) Quasi-admissible twists for set theoretic solutions.
We are focusinghenceforth on R matrices acting of tensor products of finite N dimensional spaces V, i.e. R : V ⊗ V → V ⊗ V. Our main aim is to identify the explicit formof Drinfeld’s twists for finite, involutive, non-degenerate, set theoretic solutionsof the YBE coming from braces, and also show that they are admissible. Weare employing the term quasi-admissible twist due to the fact that we only showhere the admissibility of the set theoretic twists, i.e. the validity of the co-cyclecondition, however we are not exploring the action of a co-unit on such twists.We make some preliminary remarks at the end of the manuscript illustrating thesingular nature of set theoretic twists via a simple example.More specifically, in the analysis that follows we focus on finite, involutive,non-degenerate, set theoretic solutions of the YBE given by r = P ˇ r , whereˇ r = P x,y, ∈ X e x,σ x ( y ) ⊗ e y,τ y ( x ) . We review below the fundamental constraintsemerging from the YBE for such solutions. Constraints for set theoretic solutions of the YBE.
For any finite, non-degenerate, involutive, set theoretic solution of the YBE r = P ˇ r , where ˇ r = P x,y ∈ X e x,σ x ( y ) ⊗ e y,τ y ( x ) , the following conditions hold: (3.14) σ σ η (ˆ x ) ( σ τ x ( η ) ( y )) = σ η ( σ x ( y )) and τ τ y ( x ) ( τ σ x ( y ) ( η )) = τ y ( τ x ( η )) , ∀ η, x, y ∈ X : σ η ( x ) , τ y ( x ) are fixed.Proof. It is convenient for our purposes here to express the Yang-Baxter equationin the following form:(3.15) ( I ⊗ ˇ r ) ( I ⊗ ∆) r = ( I ⊗ ∆) r ( I ⊗ ˇ r )(similarly (ˇ r ⊗ I ) (∆ ⊗ I ) r = (∆ ⊗ I ) r (ˇ r ⊗ I )). This is immediately shown byrecalling the definitions: ( I ⊗ ∆) r = r r , I ⊗ ˇ r = r , (∆ ⊗ I ) r = r r , ˇ r ⊗ I = r and ˇ r = P r. We take the following steps. (1) We first identify ( I ⊗ ∆) r = r r , where r is the set theoretic solution(3.16) ( I ⊗ ∆) r = X x,y,η ∈ X e τ y ( x ) ,σ η ( x ) ⊗ e η,τ x ( η ) ⊗ e σ x ( y ) ,y . (2) We calculate the left and right hand side of (3.15), using also the involutiveproperty, LHS = X x,y,η ∈ X e τ y ( x ) ,σ η ( x ) ⊗ e σ η ( σ x ( y )) ,τ x ( η ) ⊗ e τ σx ( y ) ( η ) ,y (3.17) RHS = X x,y,η ∈ X e τ y ( x ) ,σ η ( x ) ⊗ e η,σ τx ( η ) ( y ) ⊗ e σ x ( y ) ,τ y ( τ x ( η )) . (3.18)(3) We identify the constraints emerging from (3.15) by equating expressions(3.17) and (3.18). Equivalence of expression (3.17) and (3.18) leads to η = σ ˆ η ( σ ˆ x (ˆ y )) , σ x ( y ) = τ σ ˆ x (ˆ y ) (ˆ η ) and(3.19) τ x ( η ) = σ τ ˆ x (ˆ η ) (ˆ y ) , y = τ ˆ y ( τ ˆ x (ˆ η )) . (3.20) Note that relations (3.19) are equivalent to: σ η ( σ x ( y )) = ˆ η, and τ σ x ( y ) ( η ) = σ ˆ x (ˆ y ). We also require(3.21) σ η ( x ) = σ ˆ η (ˆ x ) and τ y ( x ) = τ ˆ y (ˆ x ) . Conditions (3.19)-(3.21) guarantee the equivalence between the LHSand RHS of (3.15) given in (3.17), (3.18).(4) We consider together (3.20), (3.21) and obtain σ η ( x ) = σ ˆ η (ˆ x ) , τ x ( η ) = σ τ ˆ x (ˆ η ) (ˆ y ), which leads to η = σ σ ˆ η (ˆ x ) ( σ τ ˆ x (ˆ η ) (ˆ y )), then comparing with (3.19)we obtain the first fundamental constraint:(3.22) C ≡ σ σ ˆ η (ˆ x ) ( σ τ ˆ x (ˆ η ) (ˆ y )) − σ ˆ η ( σ ˆ x (ˆ y )) = 0 , and conversely if C = 0 then σ η ( x ) = σ ˆ η (ˆ x ) given (3.19), (3.20).Similarly, from (3.19), (3.21) σ x ( y ) = τ σ ˆ x (ˆ y ) (ˆ η ), τ y ( x ) = τ ˆ y (ˆ x ), whichleads to y = τ τ ˆ y (ˆ x ) ( τ σ ˆ x (ˆ y ) (ˆ η )), and comparison with (3.20) leads to thesecond fundamental constraint:(3.23) C ≡ τ τ ˆ y (ˆ x ) ( τ σ ˆ x (ˆ y ) (ˆ η )) − τ ˆ y ( τ ˆ x (ˆ η )) = 0 , and conversely if C = 0 then τ ˆ y (ˆ x ) = τ y ( x ) , given (3.19), (3.20).For the brace solution of the YBE, which is our main interest in this study, theabove constraints are satisfied as shown by Rump. (cid:3) We derive in the following Lemma the explicit forms of the co-products ofthe quantum algebra associated to brace solutions of the YBE. This will be ofsignificance when identifying the corresponding admissible Drinfeld twists.
ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 13
Lemma 3.8.
Let ˇ r be an involutive, finite, non-degenerate set theoretic solutionof the YBE and ( id ⊗ ∆) r = P x,y ∈ X e x,y ⊗ ∆( L x,y ) , where L x,y are the representedelements of the corresponding quantum algebra defined by the RTT relation (2.7)and r = P ˇ r ( P is the permutation operator). Then (3.24) ∆( L x,y ) ˇ r = ˇ r ∆( L x,y ) ∀ x, y ∈ X, subject to (3.22), (3.23).Proof. The proof is an immediate consequence of the constraints emerging for settheoretic solutions of the YBE. Indeed, from (3.16) we can read of(3.25) ∆( L τ y ( x ) ,σ η ( x ) ) = X x,y,η ∈ W e η,τ x ( η ) ⊗ e σ x ( y ) ,y , where W : x, y, η ∈ X such that σ η ( x ) and τ y ( x ) are fixed .Then due to ( I ⊗ ˇ r ) (id ⊗ ∆) r = ( I ⊗ ∆) r ( I ⊗ ˇ r ) we obtain (3.24). I.e. all entriesof the ( I ⊗ ∆) r -matrix commute with ˇ r , subject to (3.22), (3.23). (cid:3) It was shown in [8] Proposition 3.3 that any brace solution of the YBE can beobtained from the permutation operator via a similarity transformation, i.e. ˇ r = F − P F
12 2 . Proposition 3.9.
Let ˇ r = P x,y ∈ X e x,σ x ( y ) ⊗ e y,τ y ( x ) be the brace solution of thebraid YBE. Let also V k , k ∈ { , . . . , N } be the eigenvectors of the permutationoperator P = P x,y ∈ X e x,y ⊗ e y,x , and ˆ V k , k ∈ { , . . . , N } be the eigenvectors ofthe brace ˇ r -matrix. Then the ˇ r -matrix can be expressed as a Drinfeld twist, suchthat ˇ r = F − PF , where the twist F − is expressed as F − = P N k =1 ˆ V k V Tk [8] . From the proof of the above Proposition (Proposition 3.3 in [8]) we know thatthe eigenvectors of the permutation operator P and the set theoretic braid solutionˇ r are given as follows. Let ˆ e k , k ∈ { , . . . , N } be the N dimensional column vectorwith 1 in the k th position and 0 elsewhere, i.e. ˆ e k form a basis of the N dimensionalvector space.(1) The (normalized) eigenvectors of the permutation operator are ( x, y ∈ X ): V k = ˆ e x ⊗ ˆ e x , k ∈ (cid:8) , . . . , N (cid:9) V k = 1 √ (cid:0) ˆ e x ⊗ ˆ e y + ˆ e y ⊗ ˆ e x (cid:1) , x = y, k ∈ (cid:8) N + 1 , . . . , N + N (cid:9) ,V k = 1 √ (cid:0) ˆ e x ⊗ ˆ e y − ˆ e y ⊗ ˆ e x (cid:1) , x = y, k ∈ (cid:8) N + N , . . . , N (cid:9) . (3.26) The first N + N eigenvectors have the same eigenvalue 1, while the rest N −N eigenvectors have eigenvalue −
1. Also it is easy to check that The twist of the present study is the inverse of F in [8]. It is just a matter of convention V k form an ortho-normal basis for the N dimensional space. Indeed, V Tk V l = δ kl and P N k =1 V k V Tk = I N ( T denotes usual transposition).(2) The eigenvectors of the ˇ r -matrix areˆ V k = ˆ e x ⊗ ˆ e y , ( x, y ) = ( σ x ( y ) , τ y ( x )) , k ∈ (cid:8) , . . . , N (cid:9) ˆ V k = 1 √ (cid:0) ˆ e x ⊗ ˆ e y + ˆ e σ x ( y ) ⊗ ˆ e τ y ( x ) (cid:1) , k ∈ (cid:8) N + 1 , . . . , N + N (cid:9) , ˆ V k = 1 √ (cid:0) ˆ e x ⊗ ˆ e y − ˆ e σ x ( y ) ⊗ ˆ e τ y ( x ) (cid:1) , ( x, y ) = ( σ x ( y ) , τ y ( x )) ,k ∈ (cid:8) N + N , . . . , N (cid:9) (3.27) As in the case of the permutation operator the ˇ r matrix has the sameeigenvalues 1 and − N + N and N −N re-spectively. Hence, the two matrices are similar, i.e. there exists some F ∈
End( V ⊗ V ) (not uniquely defined) such that ˇ r = F − PF , where F − = P N k =1 ˆ V k V Tk . From Proposition 3.3 in [8] we can extract explicit forms for the twist F andstate the following. Proposition 3.10.
Let F − = P N k =1 ˆ V k V Tk be the similarity transformation(twist) of Proposition 3.9, such that ˇ r = F − PF , where ˇ r = P x,y ∈ X e x,σ x ( y ) ⊗ e y,τ y ( x ) is the brace solution of the braid YBE, P is the permutation operator and ˆ V k , V k are their respective eigenvectors. Then the twist can be explicitly expressedas F = P x,y ∈ X e x,x ⊗ e σ x ( y ) ,y .Proof. We begin our proof by re-expressing the eigenvectors of the permutationoperator in a convenient for our purposes form. The first N eigenvectors are usedas they are, we only conveniently re-express the rest (let n = N + N ): V k = ˆ e x ⊗ ˆ e x , k ∈ (cid:8) , . . . , N (cid:9) V k = 1 √ (cid:0) ˆ e x ⊗ ˆ e σ x ( y ) + ˆ e σ x ( y ) ⊗ ˆ e x (cid:1) , x = σ x ( y ) , k ∈ (cid:8) N + 1 , . . . , n (cid:9) ,V k = 1 √ (cid:0) ˆ e x ⊗ ˆ e σ x ( y ) − ˆ e σ x ( y ) ⊗ ˆ e x (cid:1) , x = σ x ( y ) , k ∈ (cid:8) n + 1 , . . . , N (cid:9) . (3.28)Let us first mention that if x = σ x ( y ) then y = τ x ( y ) (and vice versa). Indeed,this can be show as follows: let ˆ y = τ y ( x ) , then due to involution we obtain x = σ x (ˆ y ) (and y = τ ˆ y ( x )), but due to the fact that σ x ( y ) is a bijection, andalso x = σ x ( y ) , we conclude that ˆ y = y . We may now compute F − using theexplicit expressions of the eigenvectors of ˇ r and P (3.27), (3.28) (recall n = N + N ) ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 15 x, y ∈ X : F − = N X k =1 ˆ V k V Tk + n X k = N +1 ˆ V k V Tk + N X k = n +1 ˆ V k V Tk = X x = σ x ( y ) (ˆ e x ⊗ ˆ e y )(ˆ e Tx ⊗ ˆ e Tx )+ 12 X x = σ x ( y ) (ˆ e x ⊗ ˆ e y )(ˆ e Tx ⊗ ˆ e Tσ x ( y ) ) + 12 X x = σ x ( y ) (ˆ e σ x ( y ) ⊗ ˆ e τ y ( x ) )(ˆ e Tσ x ( y ) ⊗ ˆ e Tx )= X x = σ x ( y ) e x,x ⊗ e y,x + X x = σ x ( y ) e x,x ⊗ e y,σ x ( y ) = X x,y ∈ X e x,x ⊗ e y,σ x ( y ) From the above expression and due to the fact that σ x and τ y are bijections, weconclude that F = P x,y ∈ X e x,x ⊗ e σ x ( y ) ,y , ( F − = F T , where T denotes totaltransposition in both spaces).Recall that r = P ˇ r, we also confirm by direct computation, and using the factthat σ x , τ y are bijections that ( F ( op ) ) − F = P x, ∈ X e y,σ x ( y ) ⊗ e x,τ y ( x ) = r . (cid:3) Remark 3.11.
The twist is not uniquely defined, for instance an alternative twistis of the form G = P x,y ∈ X e τ y ( x ) ,x ⊗ e y,y , and P x, ∈ X e y,σ x ( y ) ⊗ e x,τ y ( x ) = G − G .This is shown by direct computation. Remark 3.12.
The Baxterized solution of the YBE, R ( λ ) = λr + P (recall P is the permutation operator) can be also expressed as R ( λ ) = F − ˜ R ( λ ) F ,where ˜ R is the Yangian. This is a straightforward statement due to the form ofthe Yangian ˜ R ( λ ) = λ I + P , and due to the fact that F − P F = P . We now focus on the derivation of the 3-twist F and the co-cycle condition.We will now identify the 3-twist for any finite, involutive, non-degenerate settheoretic solution and show that the co-cycle condition is satisfied, i.e. F is anadmissible twist. This is achieved in the following Propositions. Proposition 3.13.
In accordance to the co-multiplication of the quantum algebraas defined in Lemma 3.8, and the notation introduced in (3.5) we define: F , = X x,y,η ∈ X e σ η ( x ) ,σ η ( x ) ⊗ e η,τ x ( η ) ⊗ e σ x ( y ) ,y | C =0 (3.29) F , = X x,y,η ∈ X e σ η ( x ) ,σ η ( x ) ⊗ e τ x ( η ) ,τ x ( η ) ⊗ e σ η ( σ x ( y )) ,y | C =0 (3.30) where C = σ σ η ( x ) ( σ τ x ( y ) ( y )) − σ η ( σ x ( y )) . Let also ˇ r = P x,y ∈ X e x,σ x ( y ) ⊗ e y,τ y ( x ) ,then (3.31) ˇ r F , = F , ˇ r , ˇ r F , = F , ˇ r . Proof.
We divide the proof in two parts.(1) We first show that ˇ r F , = F , ˇ r , using the explicit expressions of ˇ r , F , and F , : The LHS of the expression is (subject to the constraint C = 0) ˇ r F , = X η,x,y ∈ X e η,σ η ( x ) ⊗ e x,τ x ( η ) ⊗ e σ η ( σ x ( y )) ,y . Similarly, F , ˇ r = X η,x,y ∈ X e η,σ η ( x ) ⊗ e x,τ x ( η ) ⊗ e σ ση ( x ) ( σ τx ( η ) ( y )) ,y , and due to the constraint C = 0: F , ˇ r = P η,x,y ∈ X e η,σ η ( x ) ⊗ e x,τ x ( η ) ⊗ e σ η ( σ x ( y )) ,y , which concludes the first part of our proof.(2) We move on now to the second part of our proof, which is a bit moreinvolved: ˇ r F , = F , ˇ r . By direct computation given the explicitexpressions for F , and ˇ r :ˇ r F , = X η,x,y ∈ X e σ η ( x ) ,σ η ( x ) ⊗ e σ η ( σ x ( y )) ,τ x ( η ) ⊗ e τ σx ( y ) ( η ) ,y F , ˇ r = X η,x,y ∈ X e σ η ( x ) ,σ η ( x ) ⊗ e h,σ τx ( η ) ( y ) ⊗ e σ x ( y ) ,τ y ( τ x ( η )) , subject to C = 0. We now follow the logic of obtaining the constraints forset theoretic solutions of the YBE. We employ (3.19) and (3.20) which asargued due to C = 0 guarantee that σ η ( x ) = σ ˆ η (ˆ x ) leading to ˇ r F , = F , ˇ r , which basically concludes our proof.All possible permutations among the indices 0 , , (cid:3) Remark 3.14.
This is a straightforward, but useful remark. Recall, r = P ˇ r , where P is the permutation operator. If F , ˇ r = ˇ r F , , and F , ˇ r = ˇ r F , , asin Proposition 3.13, then by multiplying the latter two equalities with P from theleft we conclude: F , r = r F , , and F , r = r F , . Generalization.
The ( n + 1)-objects F , ...n := (id ⊗ ∆ ( n ) ) F and F , ...n :=(∆ ( n ) ⊗ id) F can be now derived by iteration exploiting the explicit form of T , ..n := (id ⊗ ∆ ( n ) ) r . Indeed, let us first identify T , ..n recalling the specificform of the set theoretic solution r = P x,y ∈ X e y,σ x ( y ) ⊗ e x,τ y ( x ) , (3.32) T , ..n = X x ,..x n ,y ∈ X e y ,σ xn ( y n ) ⊗ e x ,τ y ( x ) ⊗ . . . ⊗ e x n ,τ yn ( x n ) , subject to y m − = σ x m ( y m ) , m ∈ { , . . . , n } . Then according to the genericexpression (3.32) and keeping in mind that T , ..n ˇ r mm +1 = ˇ r mm +1 T , ..n , m ∈ ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 17 { , . . . , n − } (see also Proposition 3.13 and expression (3.29)) we derive, F , ...n = P x ,..x n ,y ∈ X e σ xn ( y n ) ,σ xn ( y n ) ⊗ e x ,τ y ( x ) ⊗ . . . ⊗ e x n ,τ yn ( x n ) , subject to y m − = σ x m ( y m ) , m ∈ { , . . . , n } . Also, by iteration (see expression (3.30)) we obtain, F ..n − ,n = P x ,..x n ,y ∈ X e x ,x ⊗ e x ,x ⊗ . . . e x n ,x n ⊗ e X ,y , where we define X := σ x ( σ x ( ... ( σ x n ( y )) ... ).We may now proceed in proving the admissibility of the derived twist, i.e. showthe validity of the co-cycle condition. Proposition 3.15.
Let F = F ⊗ I and F = I ⊗ F , where F = P η,x,y ∈ X e η,η ⊗ e σ η ( x ) ,x . Let also F , and F , defined in (3.29) and (3.30). Then (3.33) F := F F , = F F , . Proof.
By substituting the expressions for F , F , F , and F , (recall C = 0holds for F , , F , ) we obtain by direct computation: F F , = X η,x,y ∈ X e σ η ( x ) ,σ η ( x ) ⊗ e η,τ x ( η ) ⊗ e σ η ( σ x ( y )) ,y . Similarly, F F , = X η,x,y ∈ X e σ η ( x ) ,σ η ( x ) ⊗ e η,τ x ( η ) ⊗ e σ η ( σ x ( y )) ,y The explicit form the 3-twist is given from the expressions above as F = P η,x,y ∈ X e σ η ( x ) ,σ η ( x ) ⊗ e η,τ x ( η ) ⊗ e σ η ( σ x ( y )) ,y | C =0 . (cid:3) We are now in the position to prove the factorization of the monodromy matrix T , in terms of the admissible twists. Proposition 3.16.
Let r = P x,y ∈ X e y,σ x ( y ) ⊗ e x,τ y ( x ) , a solution of the YBE, and F = P x,y,η ∈ X e η,η ⊗ e σ η ( x ) ,x ⊗ e σ η ( σ x ( y )) ,y | C =0 , as defined in Proposition 3.15.Let also, T , = r r , then T , = F − F .Proof. To prove the decomposition of the monodromy matrix T , we employ thefollowing statements:(1) Proposition 3.7: F − n F n = r n , n ∈ { , } .(2) Proposition 3.13 and Remark 3.14: F n ,m r n = r n F n,m and F m,n r n = r n F m, n , n = m ∈ { , } .(3) Proposition 3.15: the co-cycle condition, F F , = F F , .We first recall that F n ,m = P n F n,m P n , similarly for F m,n , where P is thepermutation operator. The proof is straightforward based on the logic of theproof of Proposition 3.7: i.e. use (1)-(3) above, then F T , F − = I V ⊗ . (cid:3) Direct computation.
Let us also confirm Proposition 3.16 by direct computa-tion. We first derive T , via (3.32) for n = 2: T , = X η,x,y ∈ X e y,σ η ( σ x ( y )) ⊗ e η,τ σx ( y ) ( η ) ⊗ e x,τ y ( x ) . (3.34)We also derive F − ; due to the fact that σ y , τ x are bijective functions: F − = F T , where T denotes total transposition, i.e. transposition in all three spacesof the tensor product. We may now show by direct computation that T , = F − F , indeed F − F = (cid:16) X η,x,y ∈ X e y,σ η ( σ x ( y )) ⊗ e η,η ⊗ e x,σ η ( x ) (cid:17)(cid:16) X ˆ η, ˆ x, ˆ y ∈ X e ˆ η, ˆ η ⊗ e σ ˆ η (ˆ x ) , ˆ x ⊗ e σ ˆ η ( σ ˆ x (ˆ y )) , ˆ y (cid:17) = X x,y,η, ˆ x, ˆ y, ˆ η ∈ X e y, ˆ η ⊗ e η, ˆ x ⊗ e x, ˆ y (3.35)subject to the following conditions:(3.36) ˆ η = σ η ( σ x ( y )) , σ ˆ η (ˆ x ) = η, σ η ( x ) = σ ˆ η ( σ ˆ x (ˆ y ))Our aim now is to express the ˆ x, ˆ y, ˆ η in terms of x, y, η , notice that ˆ η is alreadyexpressed in such a way. Consider now the condition σ ˆ η (ˆ x ) = h , and let τ ˆ x (ˆ η ) = ξ then we obtain via the involutive property: ˆ η = σ η ( ξ ) and ˆ x = τ ξ ( η ). However,due to the first of the conditions (3.36) and the fact that σ η is a bijective function,we conclude that ξ = σ x ( y ) and hence ˆ x = τ σ x ( y ) ( η ). It remains now to expressˆ y in terms of x, y, η ; we consider the third of the conditions (3.36) as well ascondition C = 0 then σ η ( x ) = σ σ ˆ η (ˆ x ) ( σ τ ˆ x (ˆ η ) (ˆ y )), but as shown above σ ˆ η (ˆ x ) = h ,then using also the fact that σ η is a bijection we conclude x = σ τ ˆ x (ˆ η ) (ˆ y ). From ourconsiderations above τ ˆ x (ˆ η )(= ξ ) = σ x ( y ) , and we conclude that ˆ y = τ y ( x ).Having expressed ˆ x, ˆ y, ˆ η in terms of x, y, η : ˆ η = σ η ( σ x ( y )) , ˆ x = τ σ x ( y ) ( η ) , andˆ y = τ y ( x ) , we arrive via (3.35) at(3.37) F − F = X x,y,η ∈ X e y,σ η ( σ x ( y )) ⊗ e η,τ σx ( y ) ( η ) ⊗ e x,τ y ( x ) which is precisely T , (3.34). (cid:3) Remark 3.17.
An alternative admissible twist is derived as follows. Using thenotation introduced in (3.5), we define G , = X η,x,y ∈ X e τ y ( τ x ( η )) ,η ⊗ e σ x ( y ) ,σ x ( y ) ⊗ e τ y ( x ) ,τ y ( x ) | C =0 (3.38) G , = X η,x,y ∈ X e τ x ( η ) ,η ⊗ e y,σ x ( y ) ⊗ e τ y ( x ) ,τ y ( x ) | C =0 (3.39) C = τ y ( τ x ( η )) − τ τ y ( x ) ( τ σ x ( y ) ( η )) , then: ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 19 (1) G , ˇ r = ˇ r G , and G , ˇ r = ˇ r G , , where ˇ r is the brace solution. (2) The co-cycle condition is also satisfied and the 3-twist is then derived, i.e. G := G G , = G G , , where the 2-twist is G = P x,y ∈ X e τ y ( x ) ,x ⊗ e y,y , given in Remark 3.11. The explicit form of the 3-twist then is G = P η,x,y ∈ X e τ y ( τ x ( η )) ,h ⊗ e y,σ x ( y ) ⊗ e τ y ( x ) ,τ y ( x ) | C =0 The proofs of the statements above follow the same logic of the correspondingproofs for F . The n -fold twists may be derived by iteration using the algebra co-product rulesas identified by T , ....n = r n ...r r . Lemma 3.18.
Let F be an admissible twist, such that r = F − F , satisfies theYBE. Let also the Baxterized solutions of the YBE R ( λ ) = λr + P , ˜ R ( λ ) = λ I + P and T ( λ ) = R N ( λ ) . . . R ( λ ) , ˜T ( λ ) = R N ( λ ) . . . R ( λ ) , then (3.40) ˜T ( λ ) = F ..n T ( λ ) F − ..n . Proof.
It suffices to prove that(3.41) F ...n − n n +1 ...N R n ( λ ) = R n ( λ ) F ...n − nn +1 ...N then according to Propositions 3.5 and 3.7 expression (3.40) follows. Indeed, for theadmissible twist via Proposition 3.5, F ...n − n n +1 ...N r n = r n F ...n − nn +1 ...N , but also by the definition of the permutation operator P we have F ...n − n n +1 ...N P n = P n F ...n − nn +1 ...N . Then according to the definitionof R, ˜ R we arrive at (3.41), which concludes our proof. (cid:3) Special case: Lyubashenko’s solution.
We focus now on a special classof set theoretic solutions of the YBE known as Lyubashenko’s solutions [9] (seealso [16] in relation to symmetric groups). We first introduce this class of solutionsand we show that they can be expressed as simple twists recalling the results of[8]. We then move on to show that these are admissible twists and we explicitlyderive the associated n -twists.Let us recall Proposition 3.1 in [8]: Proposition 3.19.
Let τ, σ : X → X be isomorphisms, such that σ ( τ ( x )) = τ ( σ ( x )) = x and let V = P x ∈ X e x,τ ( x ) and V − = P x ∈ X e τ ( x ) ,x . Then anysolution of the braid YBE of the type (3.42) ˇ r = X x,y ∈ X e x,σ ( y ) ⊗ e y,τ ( x ) , can be obtained from the permutation operator P = P x,y ∈ X e x,y ⊗ e y,x as (3.43) ˇ r = ( V ⊗ I ) P ( V − ⊗ I ) = ( I ⊗ V − ) P ( I ⊗ V ) We will now derive the n -twists associated to Lyubashenko’s solution and showthat these are admissible. We introduce in what follows two distinct twists F and G compatible with the results of [8]. Let us first introduce the twist F = V , giventhe explicit from V = P x ∈ X e x,τ ( x ) we may rewrite F = P x,y ∈ X e x,x ⊗ e y,τ ( y ) . It isinstructive to compare with the general set theoretic twist, Proposition 3.10. Notethat the terms at each space now “decouple” due to the fact that σ x ( y ) , τ y ( x ) → σ ( y ) , τ ( x ).Indeed, according to Proposition 3.19, P = V ˇ r V − , where ˇ r = P x,y ∈ X e x,σ ( y ) ⊗ e y,τ ( x ) . Also, F is a Reshetikhin type twist as it trivially satisfies the YBE, andconsequently the co-product structure is the one of the quasi-triangular Hopf al-gebra:(3.44) F , = F F , F , = F F . By means of (3.44) and given that F = V : we have:(3.45) F , = V V , F , = V . The co-cycle condition is satisfied and it is nothing but the Yang-Baxter equation: F := F F , . We also derive by iteration:(3.46) F , ...n = F n . . . F , F .n, = F F . . . F n , and the explicit expressions are given as(3.47) F , ..n = V V . . . V n , F ..n, = V n . The precise form of the ( n + 1)-twist is derived in the following Lemma. Lemma 3.20.
Let F = V , be the admissible twist for Lyubashenko’s solution,then the ( n + 1) -twist is given as F ..n = Q nk =1 V kk . Proof.
The proof is straightforward given the explicit forms of F , expressions(3.47) and the generalized cocycle condition (3.8). (cid:3) Recall also that V = P x ∈ X e x,τ ( x ) , then we can express the ( n + 1)-twist as F ...n = X x,x ,...,x n ∈ X e x,x ⊗ e x ,τ ( x ) ⊗ e x ,τ ( x ) ⊗ . . . ⊗ e x n ,τ n ( x n ) . We shall also need for the next Lemma the expression F ...n = V n V n − n . . . V V . Given the explicit form of the ( n + 1)-twist from Lemma 3.20 we can showthe factorization of the monodromy matrix in a straightforward manner in thefollowing Lemma. Lemma 3.21.
Let T , ..N = r N ...r , where r is Lyubashenko’s solution and F ..N , the ( n + 1) -twist as derived in Lemma 3.20, then F − ..N F ...N = T , ...N . ET THEORETIC YBE, BRACES AND DRINFELD TWISTS 21
Proof.
The factorization can be checked by direct computation, indeed F ..N T , ..n F − ...N = (cid:16) V n V N − N . . . V V (cid:17)(cid:16) V − V N . . . V − V (cid:17)(cid:16) V nn V n − n − . . . V V (cid:17) = I ⊗ ( N +1) . (cid:3) Remark 3.22.
There is an alternative admissible twist for Lyubashenko’s solu-tions. Indeed, G = V − , is an alternative admissible, Reshetikhin type twist: (1) G , = G G and G , = G G . (2) G , ...n = V − n and G ..n, = V − V − . . . V − n . (3) Via the generalized co-cycle condition the alternative ( n + 1) -twist is ex-pressed as G ...n = Q n − k =0 V − ( n − k ) k . Recalling also that V − = P x ∈ X e x,σ ( x ) we can write G ...n = X x,x ,..,x n ∈ X e x n ,σ n ( x n ) ⊗ . . . ⊗ e x ,σ ( x ) ⊗ e x ,σ ( x ) ⊗ e x,x . Curious observations.
This is a preliminary discussion motivating the next nat-ural steps of our investigation. Throughout this manuscript we have been focusedon the identification of set theoretical twists and the issue of their admissibility inthe sense of Proposition 3.15. We have restricted our attention on the co-productstructure of the underlying quantum algebra and we have not discussed the ac-tions of the co-unit and antipode for set theoretic solutions, which are crucial inidentifying the quantum algebra as a quasi-triangular Hopf algebra.Let us briefly refer to the notions of co-unit and antipode for the quantumalgebra associated to Lyubashenko’s solution, which represents a simple, but char-acteristic example of involutive, set theoretic solution. Recall the Lyubashenkosolution can be expressed in the compact way r : V ⊗ V → V ⊗ V ( V is the N -dimvector space), such that r = V − ⊗ V , where V = P x ∈ X e x,τ ( x ) is a group likeelement. From the definition (id ⊗ ∆) r = r r and recalling the simple form ofLyubashenko’s solution we obtain ∆( V ) = V ⊗ V . We define the co-unit: ǫ ( V ) = ǫ ( V ) − = ǫ ( I ) = 1 , then it follows that ( ǫ ⊗ id)∆( V ) = (id ⊗ ǫ )∆( V ) = V . We alsodefine the antipode: s ( V ) = V − , then m (cid:0) ( s ⊗ id)∆( V ) (cid:1) = m (cid:0) (id ⊗ s )∆( V ) (cid:1) = I V . We recall the Lyubashenko twists: F = I ⊗ V and G = V − ⊗ I , we can thenreadily check, regardless the above defined action of the co-unit, that ( ǫ ⊗ id) F = V , (id ⊗ ǫ ) G = V − and via ǫ ( V ) = ǫ ( V ) − = 1 , we conclude (id ⊗ ǫ ) F = I, ( ǫ ⊗ id) G = I. Even though the conditions ( ǫ ⊗ id) F = I, (id ⊗ ǫ ) G = I are nowrelaxed, the twists F , G are still admissible in the weaker sense of Propositions3.15 and 3.16. It is also interesting to present the action of the above definedco-unit on Lyubashenkso’s r matrix: ( ǫ ⊗ id) r = V , (id ⊗ ǫ ) r = V − . This is an “uncommon” action of the co-unit, the origin of which is the fact that both the settheoretic r -matrix as well as the related twists have no semi classical analogues,i.e. they can not be expressed as formal series expansions with leading term beingthe identity.Similar observations can be made for the Baxterized solutions coming fromthe Lyubashenko r -matrix. Let ˜ R ( λ ) = λI V ⊗ V + P be the Yangian R -matrixwhere P = P x,y ∈ X e x,y ⊗ e y,x is the permutation operator and e x,y are thegenerators of gl N in the N -dimensional representation. We focus for the sakeof simplicity on the finite gl N subalgebra of the Yangian; gl N is a Hopf al-gebra with counit ǫ ( e x,y ) = 0, ǫ ( I ) = 1, antipode s ( e x,y ) = − e x,y and co-product ∆( e x,y ) = e x,y ⊗ I + I ⊗ e x,y . Moreover, ∆( e x,y ) ˜ R ( λ ) = ˜ R ( λ )∆( e x,y ) , ∀ x, y ∈ X , recalling that R ( λ ) = V ˜ R ( λ ) V − = V − ˜ R ( λ ) V , we conclude that∆ ( op ) j ( e x,y ) R ( λ ) = R ( λ )∆ j ( e x,y ) , j ∈ { , } , where the two types of twisted co-products are defined as [8]: ∆ ( e x,y ) = e x,y ⊗ I + I ⊗ e τ ( x ) ,τ ( y ) and ∆ ( e x,y ) = e σ ( x ) ,σ ( y ) ⊗ I + I ⊗ e x,y , (∆ ( e x,y ) = V ⊗ V · ∆ ( e x,y ) · V − ⊗ V − ).Given the above twisted co-products it follows that ( ǫ ⊗ id)∆ ( e x,y ) = e τ ( x ) ,τ ( y ) , (id ⊗ ǫ )∆ ( e x,y ) = e x,y and ( ǫ ⊗ id)∆ ( e x,y ) = e x,y , (id ⊗ ǫ )∆ ( e x,y ) = e σ ( x ) ,σ ( y ) i.e. the twisted co-products and co-unit do not satisfy the bialgebra axioms. Inaddition, co-associativity for the deformed co-products is also an issue, indeedit is easily shown using the explicit expressions for the co-products ∆ j : (id ⊗ ∆ )∆ ( e x,y ) = Φ − · (∆ ⊗ id)∆ ( e x,y ) · Φ and (id ⊗ ∆ )∆ ( e x,y ) = ˆΦ − · (∆ ⊗ id)∆ ( e x,y ) · ˆΦ where we define Φ = I ⊗ I ⊗ V and ˆΦ = V ⊗ I ⊗ I , i.e. the co-associativity is now reduced to an almost co-associativity. Although after applyingthe simple twists the underlying algebra is still gl N [8], strict co-associativity aswell as the axioms of the bialgebra involving the co-multiplication and co-unit arenot satisfied anymore.We have briefly demonstrated the intriguing problem of characterizing the quan-tum group associated to set theoretic solutions as a bialgebra, using the simple ex-ample of Lyubashenko’s solution. The general case, also in relation to the Yangian,as well as the associated quantum double will be addressed in detail elsewhere. Acknowledgments.
I am grateful to A. Smoktunowicz for reading the manu-script and for useful discussions and comments. Support from the EPSRC researchgrants EP/R009465/1 and EP/V008129/1 is also acknowledged.
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Dept of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS,and Maxwell Institute for Mathematical Sciences, Edinburgh
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