aa r X i v : . [ m a t h - ph ] J un HOM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS
DONALD YAU
Abstract.
We introduce a Hom-type generalization of quantum groups, called quasi-triangularHom-bialgebras. They are non-associative and non-coassociative analogues of Drinfel’d’s quasi-triangular bialgebras, in which the non-(co)associativity is controlled by a twisting map. A familyof quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular bialgebra, suchas Drinfel’d’s quantum enveloping algebras. Each quasi-triangular Hom-bialgebra comes with asolution of the quantum Hom-Yang-Baxter equation, which is a non-associative version of thequantum Yang-Baxter equation. Solutions of the Hom-Yang-Baxter equation can be obtainedfrom modules of suitable quasi-triangular Hom-bialgebras. Introduction
This paper is part of an on-going effort [63, 64, 65] to study twisted, Hom-type generalizationsof the various Yang-Baxter equations and related algebraic structures. A Hom-type generalizationof the Yang-Baxter equation [7, 8, 58], called the Hom-Yang-Baxter equation (HYBE), and itsrelationships to the braid relations and braid group representations [3, 4] were studied in [63, 64].Hom versions of the classical Yang-Baxter equation [54, 55] and Drinfel’d’s Lie bialgebras [11, 12]were studied in [65].Here we consider twisted, Hom-type generalizations of quantum groups and the quantum Yang-Baxter equation (QYBE). The quantum groups being generalized in this paper are Drinfel’d’squasi-triangular bialgebras [12]. Our generalized quantum groups (the quasi-triangular Hom-bialgebras in the title) are, in general, non-associative, non-coassociative, non-commutative, andnon-cocommutative. We also refer to these objects colloquially as
Hom-quantum groups . As we willdescribe below, suitable quasi-triangular Hom-bialgebras give rise to solutions of the HYBE.Let us first recall the definition of a quasi-triangular bialgebra and its relationships to the variousYang-Baxter equations. A quasi-triangular bialgebra ( A, R ) [12] consists of a bialgebra A and aninvertible element R ∈ A ⊗ such that the following three conditions are satisfied:∆ op ( x ) R = R ∆( x ) , (∆ ⊗ Id )( R ) = R R , ( Id ⊗ ∆)( R ) = R R . (1.0.1)Here ∆ op = τ ◦ ∆ with τ the twist isomorphism, R = R ⊗ R = 1 ⊗ R , and R = ( τ ⊗ Id ) R .The element R , called the quasi-triangular structure , satisfies the QYBE R R R = R R R . (1.0.2)Examples of quasi-triangular bialgebras include Drinfel’d’s quantum enveloping algebra U h ( g ) [12] ofa semi-simple Lie algebra or a Kac-Moody algebra g [28], the anyonic quantum groups [41], and thequantum line [39], among many others. The QYBE and quasi-triangular bialgebras are motivatedby work on the quantum inverse scattering method [14, 15, 16, 17, 18, 54, 55] and exactly solved Date : November 16, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
The Yang-Baxter equation, quasi-triangular bialgebra, quantum group, Hom-bialgebra. models in statistical mechanics [7, 8, 58]. Comprehensive expositions on quasi-triangular bialgebrascan be found, for example, in the books [10, 13, 29, 42].Quasi-triangular bialgebras and the QYBE are related to the Yang-Baxter equation as follows.Consider a module V over a quasi-triangular bialgebra ( A, R ) and the operator B : V ⊗ → V ⊗ defined as B ( v ⊗ w ) = τ ( R ( v ⊗ w )). As a consequence of the QYBE (1.0.2), the operator B satisfiesthe Yang-Baxter equation (YBE) [7, 8, 58],( B ⊗ Id )( Id ⊗ B )( B ⊗ Id ) = ( Id ⊗ B )( B ⊗ Id )( Id ⊗ B ) . (1.0.3)So one quasi-triangular bialgebra ( A, R ) gives rise to many solutions of the YBE via its modules. Forthis reason, the quasi-triangular structure R is also known as the universal R -matrix . The readermay consult [48] for discussion of different versions of the YBE and of their uses in physics.Our generalizations of quantum groups and the QYBE (as well as the HYBE) are all motivatedby Hom-Lie algebras and other Hom-type algebras. Roughly speaking, a Hom-type structure ariseswhen one strategically replaces the identity map in the defining axioms of a classical structure by ageneral twisting map α . A classical structure should be a particular example of a Hom-type structurein which the twisting map is the identity map. In particular, a Hom-Lie algebra ( L, [ − , − ] , α ) hasan anti-symmetric bracket [ − , − ] : L ⊗ → L that satisfies the Hom-Jacobi identity ,[[ x, y ] , α ( z )] + [[ z, x ] , α ( y )] + [[ y, z ] , α ( x )] = 0 , where α is an algebra self-map of the module L . Hom-Lie algebras were introduced in [22] to describethe structures on some q -deformations of the Witt and the Virasoro algebras. Earlier precursors ofHom-Lie algebras can be found in [24, 38]. Hom-Lie algebras are closely related to deformed vectorfields [2, 22, 34, 35, 36, 50, 53] and number theory [33].One can similarly define a Hom-associative algebra [43], which satisfies the
Hom-associativityaxiom ( xy ) α ( z ) = α ( x )( yz ) (Definition 2.2). So a Hom-associative algebra is not associative, but thenon-associativity is controlled by the twisting map α . One obtains a Hom-Lie algebra from a Hom-associative algebra via the commutator bracket [43, Proposition 1.7]. Conversely, the envelopingHom-associative algebra of a Hom-Lie algebra is constructed in [59] and is studied further in [61].Other papers concerning Hom-Lie algebras and related Hom-type structures are [5, 19, 20, 21, 27,44, 45, 46, 59, 60, 62, 63, 64, 65].We now describe the main results of this paper concerning Hom-type generalizations of quasi-triangular bialgebras and the QYBE. Precise definitions, statements of results, and proofs are givenin later sections.Following the patterns of Hom-Lie and Hom-associative algebras, one can define Hom-bialgebras(Definition 2.2), which are non-associative and non-coassociative generalizations of bialgebras inwhich the non-(co)associativity is controlled by the twisting map α . In section 2 we introduce quasi-triangular Hom-bialgebras (Definition 2.7), generalizing Drinfel’d’s quasi-triangular bialgebrasby strategically replacing the identity map by a twisting map α in the defining axioms. We showthat a quasi-triangular Hom-bialgebra comes equipped with a solution R of the quantum Hom-Yang-Baxter equation (QHYBE) (Theorem 2.10), which is a non-associative analogue of the QYBE. Infact, due to the non-associative nature of a Hom-bialgebra, there are two different versions of theQHYBE ((2.10.1) and (2.10.2)), both of which hold in a quasi-triangular Hom-bialgebra.In section 3 we give two general procedures by which quasi-triangular Hom-bialgebras can beconstructed. First we show that every quasi-triangular bialgebra A can be twisted into a familyof quasi-triangular Hom-bialgebras A α , where α runs through the bialgebra endomorphisms on A .Here the twisting procedure is applied to the (co)multiplication in A (Theorem 3.1). On the other OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 3 hand, if A is a quasi-triangular Hom-bialgebra with a surjective twisting map α , then we obtain asequence of quasi-triangular Hom-bialgebras by replacing R ∈ A ⊗ with ( α n ⊗ α n )( R ) for n ≥ k G (Example 3.6) and function bialgebra k ( G ) (Example 3.7) for finite abelian groups G , and Drinfel’d’s quantum enveloping algebras U h ( g )(Examples 3.8 and 3.9) for semi-simple Lie algebras g . Since U h ( g ) is non-commutative and non-cocommutative, the quasi-triangular Hom-bialgebras obtained by twisting U h ( g ) are simultaneouslynon-(co)associative and non-(co)commutative.The relationship between the QYBE (1.0.2) and the YBE (1.0.3) described above is generalized tothe Hom setting in section 4. We show that a module (suitably defined) over a quasi-triangular Hom-bialgebra with α -invariant R has a canonical solution of the HYBE given by B = τ ◦ R (Theorem 4.4).This generalizes the solution of the YBE associated to a module over a quasi-triangular bialgebra,as discussed above.We illustrate the concept of modules over a quasi-triangular Hom-bialgebra with α -invariant R insection 5. In particular, we consider the quasi-triangular Hom-bialgebra U h ( sl ) α (Example 3.9) anda sequence of modules e V n over it (Proposition 5.2). Here e V n is topologically free of rank n + 1 over C [[ h ]]. We also write down explicitly the matrix representing the canonical solution of the HYBEassociated to e V (Proposition 5.3).In the next paper in this series, we will discuss another class of Hom-quantum groups,called dual quasi-triangular (DQT) Hom-bialgebras, which are also non-(co)associative and non-(co)commutative in general. These DQT Hom-bialgebras are Hom-type generalizations of dualquasi-triangular bialgebras [23, 32, 40, 52]. In particular, they can accommodate Hom-type gener-alizations of the FRT quantum groups [49], quantum matrices M q (2), the quantum general lineargroup GL q (2), and the quantum special linear group SL q (2). Applied to the M q (2)-coaction on thequantum plane, we obtain Hom-type, non-(co)associative analogues of quantum/non-commutativegeometry. DQT Hom-bialgebras are also equipped with solutions of a Hom version of the opera-tor form of the QYBE. Solutions of the HYBE can be generated from comodules of suitable DQTHom-bialgebras. 2. Quasi-triangular Hom-bialgebras
In this section, we first recall the definition of a Hom-bialgebra (Definition 2.2). Then we definequasi-triangular Hom-bialgebras (Definition 2.7) and establish the QHYBE (Theorem 2.10). Severalcharacterizations of the axioms of a quasi-triangular Hom-bialgebra are given at the end of thissection (Theorems 2.13 and 2.14). Concrete examples of quasi-triangular Hom-bialgebras are givenin the next section.2.1.
Conventions and notations.
We work over a fixed commutative ring k of characteristic 0.Modules, tensor products, and linear maps are all taken over k . If V and W are k -modules, then τ : V ⊗ W → W ⊗ V denotes the twist isomorphism, τ ( v ⊗ w ) = w ⊗ v . For a map φ : V → W and v ∈ V , we sometimes write φ ( v ) as h φ, v i . If k is a field and V is a k -vector space, then the lineardual of V is V ∗ = Hom( V, k ). From now on, whenever the linear dual V ∗ is in sight, we tacitlyassume that k is a characteristic 0 field.Given a bilinear map µ : V ⊗ → V and elements x, y ∈ V , we often write µ ( x, y ) as xy and putin parentheses for longer products. For a map ∆ : V → V ⊗ , we use Sweedler’s notation [57] for DONALD YAU comultiplication: ∆( x ) = P ( x ) x ⊗ x . To simplify the typography in computations, we often omitthe summation sign P ( x ) . Definition 2.2. (1) A
Hom-associative algebra [43] (
A, µ, α ) consists of a k -module A , abilinear map µ : A ⊗ → A (the multiplication), and a linear self-map α : A → A such that(i) α ◦ µ = µ ◦ α ⊗ (multiplicativity) and (ii) µ ◦ ( α ⊗ µ ) = µ ◦ ( µ ⊗ α ) (Hom-associativity).(2) A Hom-coassociative coalgebra [44, 46] ( C, ∆ , α ) consists of a k -module C , a linearmap ∆ : C → C ⊗ (the comultiplication), and a linear self-map α : C → C such that (i) α ⊗ ◦ ∆ = ∆ ◦ α (comultiplicativity) and (ii) ( α ⊗ ∆) ◦ ∆ = (∆ ⊗ α ) ◦ ∆ (Hom-coassociativity).(3) A Hom-bialgebra [44, 61] is a quadruple (
A, µ, ∆ , α ) in which ( A, µ, α ) is a Hom-associativealgebra, ( A, ∆ , α ) is a Hom-coassociative coalgebra, and the following condition holds:∆ ◦ µ = µ ⊗ ◦ ( Id ⊗ τ ⊗ Id ) ◦ ∆ ⊗ . (2.2.1)The compatibility condition (2.2.1) can be restated as ∆( xy ) = P ( x )( y ) x y ⊗ x y .We will refer to α as the twisting map . In a Hom-bialgebra, the (co)multiplication is not(co)associative, but the non-(co)associativity is controlled by the twisting map α . In particular,a bialgebra is a Hom-bialgebra when equipped with α = Id . More generally, any bialgebra can betwisted into a Hom-bialgebra via a bialgebra morphism, as explained in the example below. Example 2.3. (1) If (
A, µ ) is an associative algebra and α : A → A is an algebra morphism,then A α = ( A, µ α , α ) is a Hom-associative algebra with the twisted multiplication µ α = α ◦ µ [60]. Indeed, the Hom-associativity axiom µ α ◦ ( α ⊗ µ α ) = µ α ◦ ( µ α ⊗ α ) is equal to α applied to the associativity axiom of µ . Likewise, both sides of the multiplicativity axiom α ◦ µ α = µ α ◦ α ⊗ are equal to α ◦ µ .(2) Dually, if ( C, ∆) is a coassociative coalgebra and α : C → C is a coalgebra morphism,then C α = ( C, ∆ α , α ) is a Hom-coassociative coalgebra with the twisted comultiplication∆ α = ∆ ◦ α .(3) Combining the previous two cases, if ( A, µ, ∆) is a bialgebra and α : A → A is a bialgebramorphism, then A α = ( A, µ α , ∆ α , α ) is a Hom-bialgebra. The compatibility condition (2.2.1)for ∆ α = ∆ ◦ α and µ α = α ◦ µ is straightforward to check. (cid:3) It is clear that the axioms of a Hom-coassociative coalgebra are dual to those of a Hom-associativealgebra. The following examples have to do with this duality.
Example 2.4. (1) Let ( C, ∆ , α ) be a Hom-coassociative coalgebra and C ∗ be the linear dualof C . Then we have a Hom-associative algebra ( C ∗ , ∆ ∗ , α ∗ ), where h ∆ ∗ ( φ, ψ ) , x i = h φ ⊗ ψ, ∆( x ) i and α ∗ ( φ ) = φ ◦ α (2.4.1)for all φ, ψ ∈ C ∗ and x ∈ C . This is checked in exactly the same way as for (co)associativealgebras [1, 2.1], as was done in [46, Corollary 4.12].(2) Likewise, suppose that ( A, µ, α ) is a finite dimensional Hom-associative algebra. Then( A ∗ , µ ∗ , α ∗ ) is a Hom-coassociative coalgebra, where h µ ∗ ( φ ) , x ⊗ y i = h φ, µ ( x, y ) i and α ∗ ( φ ) = φ ◦ α (2.4.2)for all φ ∈ A ∗ and x, y ∈ A [46, Corollary 4.12]. In what follows, whenever µ ∗ : A ∗ → A ∗ ⊗ A ∗ is in sight, we tacitly assume that A is finite dimensional.(3) Combining the previous two examples, suppose that ( A, µ, ∆ , α ) is a finite dimensional Hom-bialgebra. Then so is ( A ∗ , ∆ ∗ , µ ∗ , α ∗ ), where ∆ ∗ , µ ∗ , and α ∗ are defined as in (2.4.1) and(2.4.2). (cid:3) OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 5
To generalize quantum groups to the Hom setting, we need a suitably weakened notion of amultiplicative identity for Hom-associative algebras.
Definition 2.5. (1) Let (
A, µ, α ) be a Hom-associative algebra. A weak unit [20] of A is anelement c ∈ A such that α ( x ) = cx = xc for all x ∈ A . In this case, we call ( A, µ, α, c ) a weakly unital Hom-associative algebra .(2) Let (
A, µ, α, c ) be a weakly unital Hom-associative algebra and R ∈ A ⊗ . Define the follow-ing elements in A ⊗ : R = R ⊗ c, R = c ⊗ R, R = ( τ ⊗ Id )( R ) . (2.5.1)The elements in (2.5.1) are our Hom-type generalizations of the elements involved in the QYBE(1.0.2) and in the definition of a quasi-triangular bialgebra. Example 2.6 ([20] Example 2.2) . If (
A, µ,
1) is a unital associative algebra, then the Hom-associative algebra A α = ( A, µ α , α ) (Example 2.3) has a weak unit c = 1. In fact, we have µ α (1 , x ) = α ( µ (1 , x )) = α ( x ) = α ( µ ( x, µ α ( x, . So (
A, µ α , α,
1) is a weakly unital Hom-associative algebra. (cid:3)
We are now ready for the main definition of this paper, which gives a non-associative and non-coassociative version of a quasi-triangular bialgebra.
Definition 2.7. A quasi-triangular Hom-bialgebra is a tuple ( A, µ, ∆ , α, c, R ) in which( A, µ, ∆ , α ) is a Hom-bialgebra, c is a weak unit of ( A, µ, α ), and R ∈ A ⊗ satisfies the follow-ing three axioms: (∆ ⊗ α )( R ) = R R , (2.7.1a)( α ⊗ ∆)( R ) = R R , (2.7.1b)[( τ ◦ ∆)( x )] R = R ∆( x ) (2.7.1c)for all x ∈ A . The elements R , R , and R are defined in (2.5.1). Example 2.8.
A quasi-triangular bialgebra (
A, R ) (1.0.1) in the sense of Drinfel’d [12] becomesa quasi-triangular Hom-bialgebra when equipped with α = Id and c = 1. In a quasi-triangularbialgebra, it is usually assumed that the quasi-triangular structure R is invertible. In that case, theaxiom (2.7.1c) can be restated as ( τ ◦ ∆)( x ) = R ∆( x ) R − . However, in this paper, even when werefer to a quasi-triangular bialgebra (e.g., in Theorem 3.1 and Corollary 3.4), we will not have touse its counit or the invertibility of its quasi-triangular structure. See also Remark 3.2. (cid:3) Some remarks are in order.
Remark 2.9. (1) The multiplications in (2.7.1) are computed in each tensor factor using themultiplication µ in A . They all make sense because there is no iterated multiplication.(2) Since a weak unit c (Definition 2.5) is not actually a multiplicative identity, it does not makemuch sense to talk about multiplicative inverse in a weakly unital Hom-associative algebra.This is the reason for not insisting on R (in Definition 2.7) being invertible and for stating(2.7.1c) without using R − . DONALD YAU (3) Write R = P s i ⊗ t i ∈ A ⊗ . From the definitions (2.5.1) of the R ij and the requirementthat c be a weak unit, the three axioms in (2.7.1) can be restated as:(∆ ⊗ α )( R ) = X α ( s i ) ⊗ α ( s j ) ⊗ t i t j , (2.9.1a)( α ⊗ ∆)( R ) = X s j s i ⊗ α ( t i ) ⊗ α ( t j ) , (2.9.1b) X x s i ⊗ x t i = X s i x ⊗ t i x , (2.9.1c)where ∆( x ) = P x ⊗ x .A major reason for introducing quasi-triangular bialgebras ( A, R ) [12] is that the quasi-triangularstructure R satisfies the QYBE (1.0.2). As we mentioned in the introduction, the fact that R isa solution of the QYBE leads to solutions of the YBE (1.0.3) in the representations of A . Wewill generalize this relationship between the QYBE and the YBE to the Hom setting in section 4.As a first step, we now show that the element R in a quasi-triangular Hom-bialgebra satisfies twonon-associative versions of the QYBE. Theorem 2.10.
Let ( A, µ, ∆ , α, c, R ) be a quasi-triangular Hom-bialgebra. Then R satisfies thequantum Hom-Yang-Baxter equations (QHYBE) ( R R ) R = R ( R R ) (2.10.1) and R ( R R ) = ( R R ) R . (2.10.2) Proof.
First consider (2.10.1). Recall that R = c ⊗ R , and we have R R = ( Id ⊗ τ )( R R ) . (2.10.3)We compute as follows:( R R ) R = [( Id ⊗ τ )( R R )] R by (2.10.3)= [( Id ⊗ τ )( α ⊗ ∆)( R )] R by (2.7.1b)= [( α ⊗ ( τ ◦ ∆))( R )]( c ⊗ R )= ( c ⊗ R )[( α ⊗ ∆)( R )] by (2.7.1c)= R ( R R ) by (2.7.1b) . This proves that R satisfies the QHYBE (2.10.1).The other QHYBE (2.10.2) is proved by a similar computation. Since R = R ⊗ c and( τ ⊗ Id )( R R ) = R R , (2.10.5)we have: R ( R R ) = ( R ⊗ c )[(∆ ⊗ α )( R )] by (2.7.1a)= [(( τ ◦ ∆) ⊗ α )( R )]( R ⊗ c ) by (2.7.1c)= [( τ ⊗ Id ) ◦ (∆ ⊗ α )( R )] R = [( τ ⊗ Id )( R R )] R by (2.7.1a)= ( R R ) R by (2.10.5) . This finishes the proof. (cid:3)
OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 7
Remark 2.11.
Let us make it clear that the two quantum Hom-Yang-Baxter equations (2.10.1)and (2.10.2) are indeed different in general. Writing R = P s i ⊗ t i , the left-hand side in (2.10.1) is( R R ) R = ( s j s i ⊗ t j c ⊗ ct i )( c ⊗ s k ⊗ t k )= ( s j s i ⊗ α ( t j ) ⊗ α ( t i ))( c ⊗ s k ⊗ t k )= α ( s j s i ) ⊗ α ( t j ) s k ⊗ α ( t i ) t k = α ( s j ) α ( s i ) ⊗ α ( t j ) s k ⊗ α ( t i ) t k . (2.11.1)Likewise, the left-hand side in (2.10.2) is R ( R R ) = ( s j ⊗ t j ⊗ c )( s i c ⊗ cs k ⊗ t i t k )= ( s j ⊗ t j ⊗ c )( α ( s i ) ⊗ α ( s k ) ⊗ t i t k )= s j α ( s i ) ⊗ t j α ( s k ) ⊗ α ( t i t k )= s j α ( s i ) ⊗ t j α ( s k ) ⊗ α ( t i ) α ( t k ) . (2.11.2)One observes that the last lines in (2.11.1) and in (2.11.2) are different. A similar computation for R ( R R ) and ( R R ) R shows that they are different as well. Remark 2.12.
On the other hand, suppose that R is α -invariant, i.e., α ⊗ ( R ) = P α ( s i ) ⊗ α ( t i ) = R . Then one can see from (2.11.1) and (2.11.2) (and a similar computation for R ( R R ) and( R R ) R ) that the two quantum Hom-Yang-Baxter equations (2.10.1) and (2.10.2) coincide.We finish this section with some alternative characterizations of the axioms (2.7.1a) and (2.7.1b)of a quasi-triangular Hom-bialgebra. They generalize an observation in [12, p.812 (5)]. Let( A, µ, ∆ , α, c ) be a Hom-bialgebra with a weak unit c , R ∈ A ⊗ be an arbitrary element, and A ∗ be the linear dual of A . Define four linear maps λ , λ ′ , λ , λ ′ : A ∗ → A by setting λ ( φ ) = h φ ⊗ α, R i , λ ′ ( φ ) = h α ∗ ( φ ) ⊗ Id, R i ,λ ( φ ) = h α ⊗ φ, R i , λ ′ ( φ ) = h Id ⊗ α ∗ ( φ ) , R i (2.12.1)for φ ∈ A ∗ .In the following characterizations of the axioms (2.7.1a) and (2.7.1b), we use the operations∆ ∗ : A ∗ ⊗ A ∗ → A ∗ (2.4.1) and µ ∗ : A ∗ → A ∗ ⊗ A ∗ (2.4.2) (when A is finite dimensional) discussedin Example 2.4 above. Theorem 2.13.
Let ( A, µ, ∆ , α, c ) be a Hom-bialgebra with a weak unit c and R ∈ A ⊗ be anarbitrary element. With the notations (2.12.1) , the following statements are equivalent. (1) The axiom (2.7.1a) holds, i.e., (∆ ⊗ α )( R ) = R R . (2) The diagram A ∗ ⊗ A ∗ λ ′ ⊗ λ ′ / / ∆ ∗ (cid:15) (cid:15) A ⊗ A µ (cid:15) (cid:15) A ∗ λ / / A (2.13.1) is commutative.If A is finite dimensional, then the two statements above are also equivalent to the commutativity ofthe diagram A ∗ λ ′ / / µ ∗ (cid:15) (cid:15) A ∆ (cid:15) (cid:15) A ∗ ⊗ A ∗ λ ⊗ λ / / A ⊗ A. (2.13.2) DONALD YAU
Proof.
First we show the equivalence between the axiom (2.7.1a) and the commutativity of thesquare (2.13.1). Write R = P s i ⊗ t i . Axiom (2.7.1a) holds if and only if h φ ⊗ ψ ⊗ Id, (∆ ⊗ α )( R ) i = h φ ⊗ ψ ⊗ Id, R R i (2.13.3)for all φ, ψ ∈ A ∗ . The left-hand side in (2.13.3) is: h φ ⊗ ψ ⊗ Id, (∆ ⊗ α )( R ) i = h φ ⊗ ψ, ∆( s i ) i α ( t i )= h ∆ ∗ ( φ, ψ ) , s i i α ( t i )= h ∆ ∗ ( φ, ψ ) ⊗ α, s i ⊗ t i i = λ (∆ ∗ ( φ, ψ )) . On the other hand, we have R R = P α ( s i ) ⊗ α ( s j ) ⊗ t i t j (2.9.1a). So the right-hand side in(2.13.3) is: h φ ⊗ ψ ⊗ Id, R R i = h φ ⊗ ψ ⊗ Id, α ( s i ) ⊗ α ( s j ) ⊗ t i t j i = h φ, α ( s i ) ih ψ, α ( s j ) i t i t j = ( h α ∗ ( φ ) , s i i t i ) ( h α ∗ ( ψ ) , s j i t j )= ( h α ∗ ( φ ) ⊗ Id, s i ⊗ t i i ) ( h α ∗ ( ψ ) ⊗ Id, s j ⊗ t j i )= µ ( λ ′ ( φ ) , λ ′ ( ψ )) . Therefore, the condition (2.13.3) holds for all φ, ψ ∈ A ∗ if and only if the square (2.13.1) is commu-tative.Next we show the equivalence between the axiom (2.7.1a) and the commutativity of the square(2.13.2) when A is finite dimensional. The finite dimensionality of A ensures that µ ∗ is well-defined.Axiom (2.7.1a) holds if and only if h Id ⊗ Id ⊗ φ, (∆ ⊗ α )( R ) i = h Id ⊗ Id ⊗ φ, R R i (2.13.6)for all φ ∈ A ∗ . The left-hand side in (2.13.6) is: h Id ⊗ Id ⊗ φ, (∆ ⊗ α )( R ) i = ∆( s i ) h φ, α ( t i ) i = ∆( s i h α ∗ ( φ ) , t i i )= ∆( h Id ⊗ α ∗ ( φ ) , s i ⊗ t i i = ∆( λ ′ ( φ )) . Below we write µ ∗ ( φ ) = P φ ⊗ φ . The right-hand side in (2.13.6) is: h Id ⊗ Id ⊗ φ, R R i = h Id ⊗ Id ⊗ φ, α ( s i ) ⊗ α ( s j ) ⊗ t i t j i = α ( s i ) ⊗ α ( s j ) h µ ∗ ( φ ) , t i ⊗ t j i = α ( s i ) h φ , t i i ⊗ α ( s j ) h φ , t j i = h α ⊗ φ , s i ⊗ t i i ⊗ h α ⊗ φ , s j ⊗ t j i = λ ⊗ ( µ ∗ ( φ )) . Therefore, the condition (2.13.6) holds for all φ ∈ A ∗ if and only if the square (2.13.2) is commutative. (cid:3) The following result is the analogue of Theorem 2.13 for the axiom (2.7.1b).
Theorem 2.14.
Let ( A, µ, ∆ , α, c ) be a Hom-bialgebra with a weak unit c and R ∈ A ⊗ be anarbitrary element. With the notations (2.12.1) , the following statements are equivalent. (1) The axiom (2.7.1b) holds, i.e., ( α ⊗ ∆)( R ) = R R . OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 9 (2)
The diagram A ∗ ⊗ A ∗ λ ′ ⊗ λ ′ / / ∆ ∗ (cid:15) (cid:15) A ⊗ A µ op (cid:15) (cid:15) A ∗ λ / / A (2.14.1) is commutative, where µ op = µ ◦ τ .If A is finite dimensional, then the two statements above are also equivalent to the commutativity ofthe diagram A ∗ λ ′ / / µ ∗ op (cid:15) (cid:15) A ∆ (cid:15) (cid:15) A ∗ ⊗ A ∗ λ ⊗ λ / / A ⊗ A, (2.14.2) where µ ∗ op = τ ◦ µ ∗ .Proof. This proof is completely analogous to that of Theorem 2.13, so we only give a sketch. Theaxiom (2.7.1b) is equivalent to the equality h Id ⊗ φ ⊗ ψ, ( α ⊗ ∆)( R ) i = h Id ⊗ φ ⊗ ψ, R R i for all φ, ψ ∈ A ∗ . Some calculation shows that this equality is in turn equivalent to the commutativityof the square (2.14.1). To show the equivalence between (2.7.1b) and the commutativity of the square(2.14.2), one uses φ ⊗ Id ⊗ Id instead of Id ⊗ φ ⊗ ψ . (cid:3) In the special case α = Id , we have λ = λ ′ and λ = λ ′ . So in this case, the commutativediagrams (2.13.1), (2.13.2), (2.14.1), and (2.14.2) mean, respectively, that λ is an algebra morphism,that λ is a coalgebra morphism, that λ is an algebra anti-morphism, and that λ is a coalgebraanti-morphism. 3. Examples of quasi-triangular Hom-bialgebras
Before we discuss the relationships between the QHYBE ((2.10.1) and (2.10.2)) and the Homversion of the YBE, in this section we describe several classes of quasi-triangular Hom-bialgebras(Examples 3.5 - 3.9).We begin with some general twisting procedures by which quasi-triangular Hom-bialgebrascan be constructed (Theorem 3.1 - Corollary 3.4). The first twisting procedure, applied to the(co)multiplication, produces a family of quasi-triangular Hom-bialgebras from any given quasi-triangular bialgebra. Recall the definitions of a quasi-triangular Hom-bialgebra (Definition 2.7)and of a quasi-triangular bialgebra (in the paragraph containing (1.0.2)).
Theorem 3.1.
Let ( A, µ, ∆ , R ) be a quasi-triangular bialgebra and α : A → A be a bialgebra mor-phism (not-necessarily preserving ). Then A α = ( A, µ α , ∆ α , α, , R ) is a quasi-triangular Hom-bialgebra, in which µ α = α ◦ µ and ∆ α = ∆ ◦ α .Proof. As noted in Examples 2.3 and 2.6, (
A, µ α , ∆ α , α,
1) is a Hom-bialgebra with a weak unit c = 1. It remains to verify the three axioms (2.7.1) for A α . For (2.7.1a) note that, since R = P s i ⊗ t i is a quasi-triangular structure (1.0.1), we have(∆ ⊗ Id )( R ) = R R = X s i ⊗ s j ⊗ t i t j , where t i t j = µ ( t i , t j ). Also, we have ∆ α = α ⊗ ◦ ∆ because α is a bialgebra morphism. Therefore,we have (∆ α ⊗ α )( R ) = (( α ⊗ ◦ ∆) ⊗ α )( R )= α ⊗ (∆ ⊗ Id )( R )= α ⊗ ( s i ⊗ s j ⊗ t i t j )= α ( s i ) ⊗ α ( s j ) ⊗ µ α ( t i , t j ) . This proves (2.7.1a) (in the alternative formulation (2.9.1a)) for A α . Similarly, for (2.7.1b) we use( Id ⊗ ∆)( R ) = R R = X s j s i ⊗ t i ⊗ t j . This gives ( α ⊗ ∆ α )( R ) = ( α ⊗ ( α ⊗ ◦ ∆))( R )= α ⊗ ( Id ⊗ ∆) ( R )= α ⊗ ( s j s i ⊗ t i ⊗ t j )= µ α ( s j , s i ) ⊗ α ( t i ) ⊗ α ( t j ) . This proves (2.7.1b) (in the alternative formulation (2.9.1b)) for A α .For (2.7.1c) note that we have (( τ ◦ ∆)( y )) R = R ∆( y ) (1.0.1) for y ∈ A , i.e., X y s i ⊗ y t i = X s i y ⊗ t i y , where ∆( y ) = P y ⊗ y . We use it below when y = α ( x ) for x ∈ A . We have µ α (( τ ◦ ∆ α )( x ) , R ) = µ α ( α ( x ) ⊗ α ( x ) , s i ⊗ t i )= α ( α ( x ) s i ) ⊗ α ( α ( x ) t i )= α ( s i α ( x ) ) ⊗ α ( t i α ( x ) )= µ α ( R, ∆ α ( x )) . This proves (2.7.1c) for A α . (cid:3) Remark 3.2.
In the proof of Theorem 3.1, we did not use the invertibility of R . So Theorem 3.1is still true even if R is not invertible. The same goes for Corollary 3.4 below.The second twisting procedure, applied to the element R , produces a family of quasi-triangularHom-bialgebras from any given quasi-triangular Hom-bialgebra with a surjective twisting map. Theorem 3.3.
Let ( A, µ, ∆ , α, c, R ) be a quasi-triangular Hom-bialgebra with α surjective and n bea positive integer. Then A ( n ) = ( A, µ, ∆ , α, c, R α n ) is also a quasi-triangular Hom-bialgebra, where R α n = ( α n ⊗ α n )( R ) .Proof. By induction it suffices to prove the case n = 1. We need to check the three axioms (2.7.1)for R α = ( α ⊗ α )( R ) = P α ( s i ) ⊗ α ( t i ), where R = P s i ⊗ t i . Using the assumption that α is(co)multiplicative and that c is a weak unit, we compute as follows:(∆ ⊗ α )( R α ) = (∆ ⊗ α )( α ⊗ α )( R )= α ⊗ ((∆ ⊗ α )( R )) OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 11 = α ⊗ ( α ( s i ) ⊗ α ( s j ) ⊗ t i t j ) by (2.9.1a)= α ( s i ) ⊗ α ( s j ) ⊗ α ( t i t j )= α ( s i ) c ⊗ cα ( s j ) ⊗ α ( t i ) α ( t j )= ( α ( s i ) ⊗ c ⊗ α ( t i ))( c ⊗ α ( s j ) ⊗ α ( t j ))= R α R α . This proves (2.7.1a) for R α . The axiom (2.7.1b) for R α is proved similarly. Notice that we have notused the surjectivity assumption of α so far.To prove (2.7.1c) for R α , pick an element x ∈ A . Since α is assumed to be surjective, we have x = α ( y ) for some (not-necessarily unique) element y ∈ A . By the comultiplicativity of α , we have∆( x ) = X x ⊗ x = X α ( y ) ⊗ α ( y ) = X α ( y ) ⊗ α ( y ) = α ⊗ (∆( y )) . Using the multiplicativity of α , we compute as follows:[( τ ◦ ∆)( x )] R α = α ( y ) α ( s i ) ⊗ α ( y ) α ( t i )= α ⊗ ((( τ ◦ ∆)( y )) R )= α ⊗ ( R ∆( y )) by (2.7.1c)= α ( s i ) α ( y ) ⊗ α ( t i ) α ( y )= α ( s i ) x ⊗ α ( t i ) x = R α ∆( x ) . This proves (2.7.1c) for R α . (cid:3) The following result is an immediate consequence of Theorems 3.1 and 3.3.
Corollary 3.4.
Let ( A, µ, ∆ , R ) be a quasi-triangular bialgebra, α : A → A be a surjective bialgebramorphism (not-necessarily preserving ), and n be a positive integer. Then A ( n ) α = ( A, µ α , ∆ α , α, , R α n ) is a quasi-triangular Hom-bialgebra, where µ α = α ◦ µ , ∆ α = ∆ ◦ α , and R α n = ( α n ⊗ α n )( R ) . We now give a series of examples of quasi-triangular Hom-bialgebras using Theorem 3.1 andCorollary 3.4.
Example 3.5 ( Anyon-generating Hom-quantum groups ) . Consider the group bialgebra CZ /n over C generated by the cyclic group Z /n with generator g and relation g n = 1. Its comultiplicationis determined by ∆( g ) = g ⊗ g, (3.5.1)which is cocommutative. It becomes a quasi-triangular bialgebra when equipped with the non-trivialquasi-triangular structure R = 1 n n − X p,q =0 exp( − πipq/n ) g p ⊗ g q . The quasi-triangular bialgebra ( CZ /n, R ) is called the anyon-generating quantum group ([41] and[42, Example 2.1.6]) and is denoted by Z ′ /n . We can get bialgebra morphisms on CZ /n by extendingthe group morphisms α k : Z /n → Z /n, α k ( g ) = g k for k ∈ { , . . . , n − } . Moreover, α k is surjective if and only if k and n are relatively prime. By Theorem 3.1, for each k ∈ { , . . . , n − } we have a quasi-triangular Hom-bialgebra( Z ′ /n ) α k = ( CZ /n, µ α k , ∆ α k , α k , , R )with twisted (co)multiplication. Here µ α k = α k ◦ µ (with µ the multiplication in CZ /n ) and ∆ α k isdetermined by ∆ α k ( g ) = g k ⊗ g k .Suppose that k and n are relatively prime, so α k is surjective. By Corollary 3.4, for each integer t ≥ Z ′ /n ) ( t ) α k = ( CZ /n, µ α k , ∆ α k , α k , , R α t ) , where the twisted quasi-triangular structure is R α t = ( α tk ⊗ α tk )( R ) = 1 n n − X p,q =0 exp( − πipq/n ) g pk t ⊗ g qk t . (cid:3) Example 3.6 ( Hom-quantum group bialgebras ) . This is a generalization of the previous ex-ample. Let k be a characteristic 0 field, G be a finite abelian group, written multiplicatively withidentity e , and k G be its group bialgebra [1, p.58, Example 2.4]. Its comultiplication is determinedby (3.5.1) for g ∈ G . Since G is commutative, k G is both commutative and cocommutative.Identify k G ⊗ k G with k H , where H = G × G . A (not-necessarily invertible) quasi-triangularstructure on the group bialgebra k G is equivalent to a function R : G × G → k such that X xy = v R ( u, x ) R ( w, y ) = δ u,w R ( u, v ) and X xy = u R ( x, v ) R ( y, w ) = δ v,w R ( u, v ) (3.6.1)for all u, v, w ∈ G [42, Example 2.1.17], where δ u,w denotes the Kronecker delta. In fact, writing R = P u,v ∈ G R ( u, v ) u ⊗ v , the two conditions in (3.6.1) are equivalent to the axioms (∆ ⊗ Id )( R ) = R R and ( Id ⊗ ∆)( R ) = R R (1.0.1), respectively. The condition ( τ ◦ ∆( x ))( R ) = R ∆( x )is automatic because k G is both commutative and cocommutative. Fix such a quasi-triangularstructure R for the rest of this example.If α : G → G is any group morphism, then it extends naturally to a bialgebra morphism on k G ,where α ( P u c u u ) = P u c u α ( u ). By Theorem 3.1 (and Remark 3.2) we have a quasi-triangularHom-bialgebra k G α = ( k G, µ α = α ◦ µ, ∆ α = ∆ ◦ α, α, e, R ) , where µ and ∆ are the multiplication and the comultiplication in k G , respectively.Suppose, in addition, that α : G → G is a group automorphism. Then by Corollary 3.4 (andRemark 3.2) we have a quasi-triangular Hom-bialgebra k G ( n ) α = ( k G, µ α , ∆ α , α, e, R α n )for each n ≥
1. We can make the twisted quasi-triangular structure R α n = ( α n ⊗ α n )( R ) moreexplicit as follows. Thinking of R as P u,v R ( u, v ) u ⊗ v , we have α ⊗ ( R ) = X u,v R ( u, v ) α ( u ) ⊗ α ( v ) = X u,v R ( α − ( u ) , α − ( v )) u ⊗ v, since α is invertible. Therefore, R α n is equivalent to the function G × G → k given by R α n ( u, v ) = R (cid:0) ( α − ) n ( u ) , ( α − ) n ( v ) (cid:1) (3.6.2)for u, v ∈ G . (cid:3) OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 13
Example 3.7 ( Hom-quantum function bialgebras ) . This example is closely related to theprevious example. Let G be a finite abelian group, k be a characteristic 0 field, and k ( G ) be thebialgebra of functions G → k [1, p.58, Example 2.4]. Its multiplication µ is defined point-wise, i.e., h µ ( φ, ψ ) , u i = h φ, u ih ψ, u i , for u ∈ G and φ, ψ ∈ k ( G ). Its comultiplication ∆ is dual to the multiplication in G , i.e., h ∆( φ ) , ( u, v ) i = h φ, uv i for φ ∈ k ( G ) and u, v ∈ G . Since G is commutative, k ( G ) is both commutative and cocommutative.A (not-necessarily invertible) quasi-triangular structure R on the function bialgebra k ( G ) is equiv-alent to a function R : G × G → k such that R ( uv, w ) = R ( u, w ) R ( v, w ) and R ( u, vw ) = R ( u, w ) R ( u, v ) (3.7.1)for all u, v, w ∈ G [42, Example 2.1.18]. In fact, the two conditions in (3.7.1) are equivalent tothe axioms (∆ ⊗ Id )( R ) = R R and ( Id ⊗ ∆)( R ) = R R (1.0.1), respectively. The condition( τ ◦ ∆( x ))( R ) = R ∆( x ) is automatic because k ( G ) is both commutative and cocommutative. Fixsuch a quasi-triangular structure R for the rest of this example.If α : G → G is a group morphism, then it extends naturally to a bialgebra morphism α ∗ : k ( G ) → k ( G ) given by α ∗ ( φ ) = φ ◦ α . By Theorem 3.1 (and Remark 3.2) we have a quasi-triangular Hom-bialgebra k ( G ) α ∗ = ( k ( G ) , µ α ∗ = α ∗ ◦ µ, ∆ α ∗ = ∆ ◦ α ∗ , α ∗ , e, R ) . The twisted multiplication and comultiplication are given by h µ α ∗ ( φ, ψ ) , u i = h φ, α ( u ) ih ψ, α ( u ) i and h ∆ α ∗ ( φ ) , ( u, v ) i = h φ, α ( uv ) i . If the group morphism α : G → G is invertible, then so is the induced bialgebra morphism α ∗ : k ( G ) → k ( G ). By Corollary 3.4 (and Remark 3.2) we have a quasi-triangular Hom-bialgebra k ( G ) ( n ) α ∗ = ( k ( G ) , µ α ∗ , ∆ α ∗ , α ∗ , e, R α n )for each n ≥
1. As a function G × G → k , the twisted quasi-triangular structure R α n is given by R α n ( u, v ) = { (( α ∗ ) n ⊗ ( α ∗ ) n )( R ) } ( u, v ) = R ( α n ( u ) , α n ( v ))for u, v ∈ G . This is the function (3.6.2) with α and α − interchanged. (cid:3) Example 3.8 ( Hom-quantum enveloping algebras ) . Let us first recall Drinfel’d’s quantumenveloping algebra U h ( g ) [12, section 13], using the notations in [42, section 3.3]. Another expositionof U h ( g ) is given in [29, XVII]. We refer the reader to [25, 26] for discussion of semi-simple Lie algebrasand to [6, 9, 47] for basics of topological algebras over the power series algebra C [[ h ]].Let g be a finite dimensional complex semi-simple Lie algebra, A = ( a ij ) ≤ i,j ≤ n be its Cartanmatrix, { β i } ≤ i ≤ n be a system of positive simple roots, and d i = ( β i , β i ) / − , − ) is the inverse of the Killing form. Define the q -symbols: q i = e hd i / , [ m ] q i = q mi − q − mi q i − q − i , (cid:20) mr (cid:21) q i = [ m ] q i ![ r ] q i ![ m − r ] q i ! , (3.8.1)where [ r ] q i ! = [ r ] q i [ r − q i · · · [1] q i and [0] q i ! = 1.The quantum enveloping algebra U h ( g ) is defined as the topological C [[ h ]]-algebra that is topo-logically generated by the set { H i , X ± i } ≤ i ≤ n of 3 n generators with the following relations:[ H i , H j ] = 0 , [ H i , X ± j ] = ± a ij X ± j , [ X + i , X − j ] = δ ij q H i i − q − H i i q i − q − i , (3.8.2) and if i = j , − a ij X k =0 ( − k (cid:20) − a ij k (cid:21) q i X − a ij − k ± i X ± j X k ± i = 0 . (3.8.3)The comultiplication ∆ in U h ( g ) is determined by∆( H i ) = H i ⊗ ⊗ H i , ∆( X ± i ) = X ± i ⊗ q H i / i + q − H i / i ⊗ X ± i (3.8.4)for 1 ≤ i ≤ n . The bialgebra U h ( g ) becomes a quasi-triangular bialgebra when equipped with thequasi-triangular structure R = X a ∈ N n (cid:26) exp h (cid:20) t + 14 ( H a ⊗ − ⊗ H a ) (cid:21)(cid:27) P a , (3.8.5)where N is the set of non-negative integers, H a = P ni =1 a i H i for a = ( a , . . . , a n ) ∈ N n , and t = P i,j ( DA ) − ij H i ⊗ H j with D = diag( d , . . . , d n ) (i.e., the diagonal matrix with d i as its i thdiagonal entry). The symbol P a denotes a certain polynomial in the variables u i = X + i ⊗ v i = 1 ⊗ X − i that is homogeneous of degree a i in u i and v i , and P = 1 ⊗
1. More information aboutthe quasi-triangular structure R (3.8.5) can be found in [30, 37, 42, 51].We can get bialgebra morphisms on U h ( g ) as follows. Let c = ( c , . . . , c n ) ∈ C n be any n -tupleof complex numbers. Define H c = P ni =1 c i H i and the C [[ h ]]-linear map α c : U h ( g ) → U h ( g ) by α c ( u ) = e hH c ue − hH c (3.8.6)for u ∈ U h ( g ). Then α c is clearly an algebra automorphism. To see that α c is compatible with ∆,first note that α c ( H j ) = H j (3.8.7)for all j because H i commutes with H j by the first relation in (3.8.2). So (3.8.4) implies that α ⊗ c ◦ ∆and ∆ ◦ α c coincide when applied to H j . Also, we have α c ( X ± j ) = γ ± j X ± j , (3.8.8)where γ j = exp( P ni =1 c i a ij ), by the second relation in (3.8.2). (More precisely, we are using e chH i X ± j e − chH i = e ± ca ij X ± j [29, p.408 (2.5)], which is a consequence of the second relation in(3.8.2).) Since α c fixes q ± H j / j = e ± hd j H j / , we infer from (3.8.4) that α ⊗ c (∆( X ± j )) = γ ± j ∆( X ± j ) = ∆( α c ( X ± j )) . Therefore, the map α c is a bialgebra automorphism on U h ( g ). Alternatively, one can use (3.8.7)and (3.8.8) as the definition of the map α c (on the generators). Then one checks directly that α c preserves the relations (3.8.2) and (3.8.3) and is compatible with the comultiplication (3.8.4).By Theorem 3.1 and Corollary 3.4, for each n -tuple c ∈ C n and each integer t ≥
0, we have aquasi-triangular Hom-bialgebra U h ( g ) ( t ) α = ( U h ( g ) , µ α , ∆ α , α, , R α t )with α = α c (3.8.6). The twisted operations are given by µ α ( u, v ) = e hH c uve − hH c , ∆ α ( H j ) = ∆( H j ) , ∆ α ( X ± j ) = γ ± j ∆( X ± j ) = e ± P ni =1 c i a ij ∆( X ± j ) . The twisted quasi-triangular structure R α t is ( α t ⊗ α t )( R ), where α = Id . To make it moreexplicit, note that α ( H a ) = H a and α ⊗ ( t ) = t because each H j is fixed by α (3.8.7). So the OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 15 entire exponential term in R (3.8.5) is fixed by α ⊗ . For the polynomial P a , let us write it as P a ( u , . . . , u n , v , . . . , v n ). Since α ⊗ ( u j ) = α ( X + j ) ⊗ γ j ( X + j ⊗
1) = γ j u j and similarly α ⊗ ( v j ) = γ − j v j , we have( α t ⊗ α t ) P a ( u , . . . , u n , v , . . . , v n ) = P a ( γ t u , . . . , γ tn u n , γ − t v , . . . , γ − tn v n ) . Therefore, we have R α t = X a ∈ N n (cid:26) exp h (cid:20) t + 14 ( H a ⊗ − ⊗ H a ) (cid:21)(cid:27) P a ( γ t u , . . . , γ tn u n , γ − t v , . . . , γ − tn v n ) , where γ ± tj = exp( ± t P ni =1 c i a ij ). (cid:3) Example 3.9 ( Hom-quantum enveloping algebra of sl ) . Let us examine the special case g = sl of the previous example. The bialgebra U h ( sl ) was first studied in [31, 56], and its quasi-triangular structure (3.9.1) was given in [12, p.816]. Using the notations of the previous examplewith g = sl , we have n = 1, a = 2, d = 1, and U h ( sl ) is the topological C [[ h ]]-algebra generatedby { H, X ± } with relations (3.8.2), where q = q = e h/ . The relations (3.8.3) are empty for g = sl .The quasi-triangular structure is R = X a ≥ ( q − q − ) a [ a ] q ! q − a ( a +1) / (cid:26) exp h H ⊗ H + a ( H ⊗ − ⊗ H )] (cid:27) ( X a + ⊗ X a − ) . (3.9.1)As in the previous example, given any complex number c , we have a bialgebra automorphism α = α c : U h ( sl ) → U h ( sl ) defined as α ( u ) = e chH ue − chH (3.9.2)for u ∈ U h ( sl ).By Theorem 3.1 we have a quasi-triangular Hom-bialgebra U h ( sl ) α = ( U h ( sl ) , µ α , ∆ α , α, , R ) . Moreover, R (3.9.1) is α -invariant, i.e., α ⊗ ( R ) = R . This is an immediate consequence of α ( H ) = H (3.8.7) and α ( X ± ) = e ± c X ± (3.8.8). Quasi-triangular Hom-bialgebras with α -invariant R play amajor role in Theorem 4.4 below. We will revisit this example in section 5. (cid:3) Solutions of the HYBE from quasi-triangular Hom-bialgebras
In this section, we extend the relationship between the QYBE (1.0.2) and the YBE (1.0.3), asdiscussed in the introduction, to the Hom setting. Let us first recall the Hom version of the YBE.
Definition 4.1 ([63]) . Let V be a k -module and α : V → V be a linear map. The Hom-Yang-Baxter equation (HYBE) is defined as( α ⊗ B ) ◦ ( B ⊗ α ) ◦ ( α ⊗ B ) = ( B ⊗ α ) ◦ ( α ⊗ B ) ◦ ( B ⊗ α ) , (4.1.1)where B : V ⊗ → V ⊗ is a bilinear map that commutes with α ⊗ . In this case, we say that B is asolution of the HYBE for ( V, α ).The YBE (1.0.3) is the special case of the HYBE (4.1.1) in which α = Id .As in the classical case, solutions of the HYBE are closely related to the braid relations and braidgroup representations [3, 4]. Indeed, suppose that B : V ⊗ → V ⊗ is a solution of the HYBE for( V, α ). Then for n ≥ ≤ i ≤ n −
1, the operators B i = α ⊗ ( i − ⊗ B ⊗ α ⊗ ( n − i − : V ⊗ n → V ⊗ n satisfy the braid relations B i B j = B j B i if | i − j | > B i B i +1 B i = B i +1 B i B i +1 . In particular, if α and B are both invertible, then so are the operators B i . In this case, there isa corresponding representation of the braid group on V ⊗ n [63, Theorem 1.4]. Many examples ofsolutions of the HYBE can be found in [63, 64].We will show that every quasi-triangular Hom-bialgebra (in which R is fixed by α ⊗ ) gives riseto many solutions of the HYBE via its modules. To make this precise, we need a suitable notion ofmodules over a Hom-associative algebra. Definition 4.2. (1) A
Hom-module is a pair (
V, α ) consisting of a k -module V and a linearmap α .(2) Let ( A, µ, α A ) be a Hom-associative algebra (Definition 2.2). By an A -module we mean aHom-module ( M, α M ) together with a linear map λ : A ⊗ M → M such that( ab ) α M ( x ) = α A ( a )( bx ) and α M ( ax ) = α A ( a ) α M ( x ) (4.2.1)for a, b ∈ A and x ∈ M , where λ ( a, x ) is abbreviated to ax .Note that a slightly different notion of a module over a Hom-associative algebra was defined in[44]. The difference with Definition 4.2 is that in [44], the second condition in (4.2.1) is not required. Example 4.3. (1) A Hom-associative algebra (
A, µ, α ) is an A -module with structure map λ = µ . In this case, the axioms (4.2.1) are exactly the Hom-associativity and the multiplicativityof α in A .(2) Let ( A, µ ) be an associative algebra, M be an A -module (in the usual sense) with structuremap λ , α A : A → A be an algebra morphism, and α M : M → M be a linear map. Supposethat α M ◦ λ = λ ◦ ( α A ⊗ α M ). This is the case, for example, if α A = Id and α M is an A -module morphism. Define the twisted action λ α = α M ◦ λ . Then it is easy to check that( M, α M ) becomes a module over the Hom-associative algebra A α = ( A, µ α = α A ◦ µ, α A )(Example 2.3) with structure map λ α . (cid:3) In a quasi-triangular Hom-bialgebra (Definition 2.7), we say that the element R is α -invariant if α ⊗ ( R ) = R . Some examples of α -invariant R were given in Example 3.9. When R is α -invariant,the two versions of the QHYBE ((2.10.1) and (2.10.2)) coincide, as was noted in Remark 2.12.We can now describe the relationship between quasi-triangular Hom-bialgebras and the HYBE(4.1.1). Theorem 4.4.
Let ( A, µ, ∆ , α, c, R ) be a quasi-triangular Hom-bialgebra in which R is α -invariantand ( M, α M ) be an A -module. Then the operator B = τ ◦ R : M ⊗ → M ⊗ is a solution of the HYBE for ( M, α M ) .Proof. To simplify the typography, we will omit the subscripts in α A and α M . Write R = P s i ⊗ t i .Then the α -invariance of R means that X s i ⊗ t i = X α ( s i ) ⊗ α ( t i ) . (4.4.1)Recall that τ denotes the twist isomorphism. So the map B = τ ◦ R is given by B ( v ⊗ w ) = X t i w ⊗ s i v OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 17 for v, w ∈ M . That B commutes with α ⊗ follows from the α -invariance of R and the second axiomin (4.2.1).It remains to check that B = τ ◦ R satisfies the HYBE (4.1.1). Note that the α -invariance of R (4.4.1) and the computation (2.11.1) imply that the QHYBE (2.10.1) now takes the form X s j s i ⊗ t j s k ⊗ t i t k = ( R R ) R = R ( R R ) = X s j s i ⊗ s k t i ⊗ t k t j . (4.4.2)Let x denote a typical generator u ⊗ v ⊗ w ∈ M ⊗ . Using the α -invariance of R (4.4.1) and themodule axioms (4.2.1), a direct computation gives:( B ⊗ α )( α ⊗ B )( B ⊗ α )( x ) = t k ( t j α ( w )) ⊗ s k α ( t i v ) ⊗ α ( s j ( s i u ))= t k ( t j α ( w )) ⊗ s k ( α ( t i ) α ( v )) ⊗ α ( s j )( α ( s i ) α ( u ))= α ( t k )( t j α ( w )) ⊗ α ( s k )( t i α ( v )) ⊗ α ( s j )( s i α ( u ))= ( t k t j ) α ( w ) ⊗ ( s k t i ) α ( v ) ⊗ ( s j s i ) α ( u )= ( t i t k ) α ( w ) ⊗ ( t j s k ) α ( v ) ⊗ ( s j s i ) α ( u ) by (4.4.2)= α ( t i )( t k α ( w )) ⊗ α ( t j )( s k α ( v )) ⊗ α ( s j )( s i α ( u ))= α ( t i )( α ( t k ) α ( w )) ⊗ t j ( α ( s k ) α ( v )) ⊗ s j ( s i α ( u ))= α ( t i ( t k w )) ⊗ t j α ( s k v ) ⊗ s j ( s i α ( u ))= ( α ⊗ B )( B ⊗ α )( α ⊗ B )( x ) . This proves that B = τ ◦ R is a solution of the HYBE for ( M, α M ). (cid:3) Corollary 4.5.
Let ( A, µ, ∆ , R ) be a quasi-triangular bialgebra, M be an A -module with structuremap λ , α A : A → A be a bialgebra morphism such that α ⊗ A ( R ) = R , and α M : M → M be a linearmap such that α M ◦ λ = λ ◦ ( α A ⊗ α M ) . Then the operator B α : M ⊗ → M ⊗ defined by B α ( v ⊗ w ) = X α M ( λ ( t i , w )) ⊗ α M ( λ ( s i , v )) (4.5.1) for v, w ∈ M , where R = P s i ⊗ t i , is a solution of the HYBE for ( M, α M ) .Proof. By Theorem 3.1 A α = ( A, µ α , ∆ α , α, , R ) is a quasi-triangular Hom-bialgebra, and ( M, α M )is an A α -module (Example 4.3) with structure map λ α = α M ◦ λ . Therefore, Theorem 4.4 impliesthat there is a solution of the HYBE for ( M, α M ) of the form B α ( v ⊗ w ) = X λ α ( t i , w ) ⊗ λ α ( s i , v ) = X α M ( λ ( t i , w )) ⊗ α M ( λ ( s i , v )) , as was to be shown. (cid:3) The following result is the special case of the previous Corollary with α A = Id . Corollary 4.6.
Let ( A, µ, ∆ , R ) be a quasi-triangular bialgebra, M be an A -module, and α M be an A -module morphism. Then the operator B α : M ⊗ → M ⊗ defined in (4.5.1) is a solution of theHYBE for ( M, α M ) . Modules over U h ( sl ) α In this section, we illustrate the results in the previous section with certain modules over the quasi-triangular Hom-bialgebra U h ( sl ) α , which was discussed in Example 3.9. We use the same notationsas in Examples 3.8 and 3.9. In particular, U h ( sl ) is the topological C [[ h ]]-algebra generated by { H, X ± } with relations (3.8.2), where q = q = e h/ , and its comultiplication is defined as in (3.8.4).It becomes a quasi-triangular Hom-bialgebra when equipped with the quasi-triangular structure R (3.9.1). Fix a complex number c , and let α : U h ( sl ) → U h ( sl ) be the bialgebra automorphism defined by α ( H ) = H and α ( X ± ) = γ ± X ± , where γ = e c . Equivalently, α is the inner automorphism (3.9.2)induced by e chH . Then U h ( sl ) α is the quasi-triangular Hom-bialgebra obtained from U h ( sl ) bytwisting its (co)multiplication along α (Theorem 3.1). Moreover, the element R (3.9.1) is α -invariant,i.e., ( α ⊗ α )( R ) = R .Fix a non-negative integer n . Let e V n be the free C [[ h ]]-module with a basis { v i } ≤ i ≤ n . Then e V n becomes a (topological) U h ( sl )-module via the map ρ : U h ( sl ) ⊗ e V n → e V n determined by ρ ( X + , v i ) = [ n + 1 − i ] q v i − , ρ ( X − , v i ) = [ i + 1] q v i +1 , ρ ( H, v i ) = ( n − i ) v i (5.0.1)for 0 ≤ i ≤ n . In (5.0.1) we set v − = 0 = v n +1 , and [ m ] q is defined as in (3.8.1). See, for example,[29, XVII.4] and [12, 31, 56]. We will apply Corollary 4.5 to the U h ( sl )-module e V n .Consider the C [[ h ]]-linear automorphism α : e V n → e V n defined by α ( v i ) = γ − i v i (5.0.2)for 0 ≤ i ≤ n . Lemma 5.1.
We have α ◦ ρ = ρ ◦ ( α ⊗ α ) (5.1.1) as maps U h ( sl ) ⊗ e V n → e V n .Proof. It suffices to check (5.1.1) on the elements X ± ⊗ v i and H ⊗ v i . When applied to H ⊗ v i ,both sides of (5.1.1) are equal to ( n − i ) γ − i v i . On the other hand, we have α ( ρ ( X + , v i )) = α ([ n + 1 − i ] q v i − )= [ n + 1 − i ] q γ − i +1 v i − = ρ ( γX + , γ − i v i )= ρ ( α ( X + ) , α ( v i )) . A similar computation shows that both sides of (5.1.1), when applied to X − ⊗ v i , are equal to[ i + 1] q γ − i − v i +1 . (cid:3) Proposition 5.2.
The map ρ α = α ◦ ρ (5.1.1) gives ( e V n , α ) the structure of a U h ( sl ) α -module.Proof. This is an immediate consequence of Lemma 5.1 and Example 4.3 (part (2)). (cid:3)
Moreover, by Lemma 5.1 and Corollary 4.5, there is a solution of the HYBE for ( e V n , α ) of theform (4.5.1): B α = α ⊗ ◦ τ ◦ R : e V ⊗ n → e V ⊗ n , (5.2.1)where R (3.9.1) acts on e V ⊗ n via the original U h ( sl )-module structure ρ . Let us write down B α explicitly for the case e V . Proposition 5.3.
With respect to the basis { v ⊗ v , v ⊗ v , v ⊗ v , v ⊗ v } of e V ⊗ , the solution B α = α ⊗ ◦ τ ◦ R of the HYBE for ( e V , α ) is given by the matrix B α = q − q γ − γ − γ − ( q − q − ) 00 0 0 qγ − , (5.3.1) where q = e h/ . OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 19
Proof.
Let us first compute the action of R (3.9.1) on e V ⊗ . It follows from the definition (5.0.1)that, when n = 1, both X a + and X a − act trivially on e V for a ≥
2, and Hv i = ρ ( H, v i ) = ( − i v i (5.3.2)for i = 0 ,
1. Thus, only the first two terms in R (3.9.1) (corresponding to a = 0 ,
1) can act non-trivially on e V ⊗ . Since q = e h/ , these two terms are R ′ = q ( H ⊗ H ) + (1 − q − ) q ( H ⊗ H + H ⊗ − ⊗ H ) ( X + ⊗ X − ) . (5.3.3)Consider the action of the first term in R ′ . It follows from (5.3.2) that H m v = v and H m v =( − m v for m ≥
0. Thus, we have q ( H ⊗ H ) = e h ( H ⊗ H ) : v ⊗ v q v ⊗ v ,v ⊗ v q − v ⊗ v ,v ⊗ v q − v ⊗ v ,v ⊗ v q v ⊗ v . (5.3.4)For the second term in R ′ , note that X + ⊗ X − only acts non-trivially on v ⊗ v among the fourbasis elements { v i ⊗ v j } ≤ i,j ≤ . We have( X + ⊗ X − )( v ⊗ v ) = v ⊗ v (5.3.5)and q ( H ⊗ H + H ⊗ − ⊗ H ) = e h ( H ⊗ H + H ⊗ − ⊗ H ) : v ⊗ v q v ⊗ v . (5.3.6)The result now follows from (5.0.2), (5.2.1), and (5.3.3) - (5.3.6). (cid:3) Regard γ = e c as a parameter, where c runs through the complex numbers. The operators B α (5.3.1) thus form a 1-parameter family of deformations of B = q − q q − q −
00 0 0 q , which is a solution of the YBE (1.0.3) for e V . References [1] E. Abe, Hopf algebras, Cambridge Tracts in Math. 74, Cambridge Univ. Press, Cambridge, UK, 1977.[2] F. Ammar and A. Makhlouf, Hom-Lie superalgebras and Hom-Lie admissible superalgebras, arXiv:0906.1668v1.[3] E. Artin, Theorie der Z¨opfe, Abh. Math. Sem. Univ. Hamburg 4 (1925) 47-72.[4] E. Artin, Theory of braids, Ann. Math. (2) 48 (1947) 101-126.[5] H.Ataguema, A. Makhlouf, and S. Silvestrov, Generalization of n -ary Nambu algebras and beyond,arXiv:0812.4058v1.[6] M.F. Atiyah and I.G. MacDonald, Introduction to commutative algebra, Addison-Wesley, Massachusetts, 1969.[7] R.J. Baxter, Partition function for the eight-vertex lattice model, Ann. Physics 70 (1972) 193-228.[8] R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London, 1982.[9] N. Bourbaki, Alg`ebre Commutative, Hermann, Paris, 1961.[10] V. Chari and A.N. Pressley, A guide to quantum groups, Cambridge Univ. Press, Cambridge, 1994.[11] V.G. Drinfel’d, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classicalYang-Baxter equations, Sov. Math. Dokl. 268 (1983) 285-287.[12] V.G. Drinfel’d, Quantum groups, in: Proc. ICM (Berkeley, 1986), p.798-820, AMS, Providence, RI, 1987.[13] P. Etingof and O. Schiffmann, Lectures on quantum groups, 2nd ed., Int. Press of Boston, Cambridge, 2002.[14] L.D. Faddeev, Quantum completely integrable models in field theory, Soviet Sci. Reviews Sec. C 1 (1980)107-155; Harwood Acad. Pub., Chur (Switzerland)-New York. [15] L.D. Faddeev, Integrble models in (1 + 1)-dimensional quantum field theory, Lectures at Les Houches 1982,Elsevier, New York, 1984.[16] L.D. Faddeev and E.K. Sklyanin, The quantum-mechanical approach to completely integrable models of fieldtheory, Dokl. Akad. Nauk SSSR 243 (1978) 1430-1433.[17] L.D. Faddeev, E.K. Sklyanin, and L.A. Takhtajan, The quantum inverse problem I, Theoret. Math. Phys. 40(1979) 194-220.[18] L.D. Faddeev and L.A. Takhtajan, The quantum inverse problem method and the XYZ Heisenberg model,Uspekhi Mat. Nauk 34 (1979) 13-63.[19] Y. Fr´egier and A. Gohr, On Hom type algebras, arXiv:0903.3393v1.[20] Y. Fr´egier and A. Gohr, On unitality conditions for hom-associative algebras, arXiv:0904.4874v1.[21] A. Gohr, On hom-algebras with surjective twisting, arXiv:0906.3270.[22] J.T. Hartwig, D. Larsson, and S.D. Silvestrov, Deformations of Lie algebras using σ -derivations, J. Algebra 295(2006) 314-361.[23] T. Hayashi, Quantum groups and quantum determinants, J. Algebra 152 (1992) 146-165.[24] N. Hu, q -Witt algebras, q -Lie algebras, q -holomorph structure and representations, Alg. Colloq. 6 (1999) 51-70.[25] J.E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math. 9, Springer, NewYork, 1972.[26] N. Jacobson, Lie algebras, Dover Pub., New York, 1962.[27] Q. Jin and X. Li, Hom-Lie algebra structures on semi-simple Lie algebras, J. Algebra 319 (2008) 1398-1408.[28] V.G. Kac, Infinite dimensional Lie algebras, Birkh¨auser, Boston, 1983.[29] C. Kassel, Quantum groups, Grad. Texts in Math. 155, Springer-Verlag, New York, 1995.[30] A.N. Kirillov and N.Yu. Reshetikhin, q -Weyl group and a multiplicative formula for univeral R sl ( F OM-QUANTUM GROUPS I: QUASI-TRIANGULAR HOM-BIALGEBRAS 21 [50] L. Richard and S. Silvestrov, A note on quasi-Lie and Hom-Lie structures of σ -derivations of C [ z ± , . . . , z ± n ],in: S. Silvestrov et. al. eds., Gen. Lie theory in Math., Phys. and Beyond, Ch. 22, pp. 257-262, Springer-Verlag,Berlin, 2009.[51] M. Rosso, An analogue of P.B.W. theorem and the universal R -matrix for U h sl ( N + 1), Comm. Math. Phys.124 (1989) 307-318.[52] P. Schauenburg, On coquasitriangular Hopf algebras and the quantum Yang-Baxter equation, Algebra Berichte67, Verlag Reinhard Fischer, M¨unchen, 1992.[53] G. Sigurdsson and S. Silvestrov, Lie color and Hom-Lie algebras of Witt type and their central extensions, in:S. Silvestrov et. al. eds., Gen. Lie theory in Math., Phys. and Beyond, Ch. 21, pp. 247-255, Springer-Verlag,Berlin, 2009.[54] E.K. Sklyanin, On complete integrability of the Landau-Lifshitz equation, LOMI preprint E-3-1979, Leningrad,1979.[55] E.K. Sklyanin, The quantum version of the inverse scattering method, Zap. Nauchn. Sem. LOMI 95 (1980)55-128.[56] E.K. Sklyanin, On an algebra generated by quadratic relations, Uspekhi Mat. Nauk 40:2 (242) (1985) 214.[57] M.E. Sweedler, Hopf algebras, Benjamin, New York, 1969.[58] C.N. Yang, Some exact results for the many-body problem in one dimension with replusive delta-functioninteraction, Phys. Rev. Lett. 19 (1967) 1312-1315.[59] D. Yau, Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl. 2 (2008) 95-108.[60] D. Yau, Hom-algebras and homology, arXiv:0712.3515v2.[61] D. Yau, Hom-bialgebras and comodule algebras, arXiv:0810.4866.[62] D. Yau, Module Hom-algebras, arXiv:0812.4695v1.[63] D. Yau, The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras, J. Phys. A 42(2009) 165202 (12pp).[64] D. Yau, The Hom-Yang-Baxter equation and Hom-Lie algebras, arXiv:0905.1887.[65] D. Yau, The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras, arXiv:0905.1890. Department of Mathematics, The Ohio State University at Newark, 1179 University Drive, Newark,OH 43055, USA
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