Quantum double construction for graded Hopf algebras

1993 ◽  
Vol 47 (3) ◽  
pp. 353-375 ◽  
Author(s):  
M.D. Gould ◽  
R.B. Zhang ◽  
A.J. Bracken
Keyword(s):  
Explicit Formula ◽  
Hopf Algebras ◽  
Quantum Double ◽  
Detailed Proof

A detailed proof of the quantum double construction is given for Z2 -graded Hopf algebras, and an explicit formula for the graded universal R−matrix is obtained in a general fashion.

scholarly journals A quantum double construction in Rel

2012 ◽  
Vol 22 (4) ◽  
pp. 618-650 ◽  
Author(s):  
MASAHITO HASEGAWA
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Binary Relations ◽  
Monoidal Categories ◽  
Quantum Double ◽  
Closed Category ◽  
Category Of Modules ◽  
Extra Structure

We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

scholarly journals Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

1997 ◽  
Vol 187 (1) ◽  
pp. 159-200 ◽  
Author(s):  
Florian Nill ◽  
Kornél Szlachányi
Keyword(s):  
Hopf Algebras ◽  
Quantum Double

HOPF-ALGEBRAIC ASPECTS OF ITERATED STOCHASTIC INTEGRALS

2009 ◽  
Vol 12 (03) ◽  
pp. 479-496 ◽  
Author(s):  
R. L. HUDSON
Keyword(s):  
Hopf Algebra ◽  
Explicit Formula ◽  
Hopf Algebras ◽  
Stochastic Integrals ◽  
Algebra Structure ◽  
Hopf Algebra Structure

An explicit formula is found for the antipode in Itô–Hopf algebras. The Hopf algebra structure is extended to iterated classical and quantum stochastic integrals. The antipode is related to change of orientation in the simplicial domains of such integrals.

scholarly journals COQUASITRIANGULAR STRUCTURES FOR EXTENSIONS OF HOPF ALGEBRAS. APPLICATIONS

2012 ◽  
Vol 55 (1) ◽  
pp. 201-215 ◽  
Author(s):  
A. L. AGORE
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Double ◽  
Bijective Correspondence ◽  
Infinite Dimensional ◽  
Main Application ◽  
Split Extensions ◽  
Necessary And Sufficient

AbstractLet A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

COREPRESENTATION THEORY OF MULTIPLIER HOPF ALGEBRAS II

2000 ◽  
Vol 11 (02) ◽  
pp. 233-278 ◽  
Author(s):  
HIDEKI KUROSE ◽  
ALFONS VAN DAELE ◽  
YINHUO ZHANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Covariant Theory ◽  
Crossed Products ◽  
Quantum Double ◽  
Multiplier Algebra ◽  
Algebraic Quantum ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Invertible Elements

We continue our development of the corepresentation theory of multiplier Hopf algebras. In this paper, we consider the corepresentations of a multiplier Hopf algebra A in a nondegenerate algebra B rather than on a vector space (cf. [25]). We concentrate ourself on those corepresentations of A in B which are invertible elements of the multiplier algebra M(B⊗A). They are called the unitary corepresentations of A. In particular, the generalized R-matrices or quasi-triangular structures of a regular multiplier Hopf algebra are unitary (bi)corepresentations. As an application the quantum double of an algebraic quantum group can be constructed by means of the universal unitary corepresentation. Moreover, a unitary corepresentation of A in B can implement an inner coaction of A on B which allows us to study the covariant theory and crossed products.

TWISTED DRINFELD DOUBLES AND REPRESENTATIONS OF A HOPF ALGEBRA

2012 ◽  
Vol 11 (06) ◽  
pp. 1250118
Author(s):  
LI BAI ◽  
SHUANHONG WANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Direct Proof ◽  
Irreducible Representations ◽  
Quantum Double ◽  
Drinfeld Double ◽  
Braided Group ◽  
R Matrix ◽  
Coalgebra Structure

In this paper, we consider a class of non-commutative and non-cocommutative Hopf algebras Hp(α, q, m) and then show that these Hopf algebras can be realized as a quantum double of certain Hopf algebras with respect to Hopf skew pairings (Ap(q, m), Bp(q, m), τα). Furthermore, using the Hopf skew pairing with appropriate group homomorphisms: ϕ : π → Aut (Ap(q, m)) and ψ : π → Aut (Bp(q, m)), we construct a twisted Drinfeld double D(Ap(q, m), Bp(q, m), τ; ϕ, ψ) which is a Turaev [Formula: see text]-coalgebra, where the group [Formula: see text] is a twisted semi-direct square of a group π. Then we obtain its quasi-triangular Turaev [Formula: see text]-coalgebra structure. We also study irreducible representations of Hp(1, q, m) and construct a corresponding R-matrix. Finally, we introduce the notion of a left Yetter–Drinfeld category over a Turaev group coalgebra and show that such a category is a Turaev braided group category by a direct proof, without center construction. As an application, we consider the case of the quasi-triangular Turaev [Formula: see text]-coalgebra structure on our twisted Drinfeld double.

Twisting products of Hopf algebras and quantum double

1999 ◽  
Vol 44 (4) ◽  
pp. 309-312
Author(s):  
Guoquan Hu
Keyword(s):  
Hopf Algebras ◽  
Quantum Double
Sign in / Sign up

Export Citation Format

Share Document