Algebraic Aspects of the Quantum Yang-Baxter Equation

Lambe, Larry A.; Radford, David E.

doi:10.1006/jabr.1993.1014

Cited by 78 publications

(46 citation statements)

References 29 publications

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“…The Yang-Baxter solutions of Theorems 2.1 and 2.2 are closely related to solutions of the quantum Yang-Baxter equation studied by Lambe, Radford, and Yetter [3,8,11]. We give the details of this relationship for 2.2; there is a similar statement for 2.1.…”

Section: Then C Is a Solution Of The Yang-baxter Equation If The Antmentioning

confidence: 79%

Higher Conjugations and the Yang–Baxter Equation

Whitehouse¹

2002

Journal of Algebra

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Let H be a Hopf algebra with invertible antipode. Two different actions of the braid group B n on H ⊗n are described. These are representations of the braid group associated to solutions of the Yang-Baxter equation. By passing to a submodule for one action and to a quotient module for the other, we obtain two actions of B n on H ⊗n−1 . In this way we recover the actions described by the author in [Comm. Algebra 29 (2001)

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Section: Then C Is a Solution Of The Yang-baxter Equation If The Antmentioning

confidence: 79%

Higher Conjugations and the Yang–Baxter Equation

Whitehouse¹

2002

Journal of Algebra

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“…Since n is a comodule map, it follows that p(n)-n®D e ker(n®Ic) = N ®C. Therefore B -{m, n) is a basis for Af such that (5) p(m) = m®A and p(n) = m ® X + n ®D for some X e C. Applying the comodule axioms to m and n we see that (6) A, D are grouplike elements,…”

Section: When Ated(r) Is Generated By Grouplike Elementsmentioning

confidence: 94%

“…Then (Af, •, p) is called a left crossed f/cop-bimodule [15], or a left quantum Yang-Baxter Hmodule [6,12]…”

Section: Preliminariesmentioning

confidence: 99%

Solutions to the Quantum Yang-Baxter Equation Arising from Pointed Bialgebras

Radford¹

1994

Transactions of the American Mathematical Society

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Abstract. Let R: M ® M -* M ® M be a solution to the quantum YangBaxter equation, where M is a finite-dimensional vector space over a field k . We introduce a quotient Anä(R) of the bialgebra A(R) constructed by Fadeev, Reshetihkin and Takhtajan, whose characteristics seem to more faithfully reflect properties R possesses as a linear operator. We characterize all R such that Ared(R) is a pointed bialgebra, and we determine all solutions R to the quantum Yang-Baxter equation when Ared(R) is pointed and dim M = 2 (with a few technical exceptions when k has characteristic 2). Extensions of such solutions to the quantum plane are studied.

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“…For all unexplained terminology we refer to [8,24]. Throughout this paper H is a Hopf algebra with bijective antipode S. Then H op and H cop are also Hopf algebras with antipode S −1 [8,15,23,27]), sometimes also called an H-crossed module or a quantum Yang-Baxter H-module, is a k-module M which is at once a left H-module and a right Hcomodule satisfying the compatibility relation…”

Section: Preliminariesmentioning

confidence: 99%

The Brauer group of Yetter-Drinfel’d module algebras

Caenepeel¹,

Oystaeyen

Zhang³

1997

Trans. Amer. Math. Soc.

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Abstract. Let H be a Hopf algebra with bijective antipode. In a previous paper, we introduced H-Azumaya Yetter-Drinfel d module algebras, and the Brauer group BQ(k, H) classifying them. We continue our study of BQ(k, H), and we generalize some properties that were previously known for the BrauerLong group. We also investigate separability properties for H-Azumaya algebras, and this leads to the notion of strongly separable H-Azumaya algebra, and to a new subgroup of the Brauer group BQ(k, H). IntroductionLet k be a commutative ring. Wall [26] introduced a Brauer group of Z/2Z-graded algebras. During the early seventies, some generalizations to gradings by other groups have been proposed, cf. e.g. [9]. In [14], Long introduced a Brauer group of H-dimodule algebras (that is, algebras with an H-action and an H-coaction), where H is a finitely generated, projective, commutative and cocommutative Hopf algebra. Long's Brauer group, today usually referred to as the Brauer-Long group, turned out to be the appropriate generalization of the BrauerWall group, since it contains all previously proposed generalizations as subgroups.Several authors have investigated the properties of the Brauer-Long group, cf. e.g. [1,2,5,6,7,10,19,20] (this list is not exhaustive). In all computations, the fact that the basic Hopf algebra is finitely generated, projective, commutative and cocommutative seems to be essential. This is a severe restriction, since, apart from finite abelian group rings, examples of such Hopf algebras are scarce. In a previous paper [8], we have been able to generalize Long's construction to arbitrary Hopf algebras-the only remaining condition is that the Hopf algebra H needs to have a bijective antipode. The underlying philosophy is that we replace Long's dimodules and dimodule algebras by Yetter-Drinfel d modules (also called quantum YangBaxter modules or crossed modules) and Yetter-Drinfel d module algebras. The Brauer group BQ(k, H) then consists of Brauer equivalence classes of H-Azumaya algebras ; if H is finitely generated, projective, commutative and cocommutative, then BQ(k, H) is anti-isomorphic to the Brauer-Long group BD(k, H). The fact that we have an anti-isomorphism rather than an isomorphism comes from the

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Algebraic Aspects of the Quantum Yang-Baxter Equation

Cited by 78 publications

References 29 publications

Higher Conjugations and the Yang–Baxter Equation

Higher Conjugations and the Yang–Baxter Equation

Solutions to the Quantum Yang-Baxter Equation Arising from Pointed Bialgebras

The Brauer group of Yetter-Drinfel’d module algebras

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