Double-bosonization of braided groups and the construction of Uq(g)

1999 ◽  
Vol 125 (1) ◽  
pp. 151-192 ◽  
Author(s):  
S. MAJID
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Cartan Subalgebra ◽  
Positive Root ◽  
Point Of View ◽  
Root Space ◽  
Quantum Double ◽  
Braided Group ◽  
Negative Root ◽  
Quasitriangular Hopf Algebra

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.

scholarly journals On the Structure of Monodromy Algebras and Drinfeld Doubles

1997 ◽  
Vol 09 (03) ◽  
pp. 371-395
Author(s):  
Florian Nill
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Tensor Products ◽  
Spin Chains ◽  
Coadjoint Action ◽  
Braided Group ◽  
Gauge Transformations ◽  
Loop Algebras ◽  
Algebraic Tensor Product ◽  
Quasitriangular Hopf Algebra

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double [Formula: see text]. Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

TOPOLOGICAL QUANTUM DOUBLE

1994 ◽  
Vol 06 (02) ◽  
pp. 305-318 ◽  
Author(s):  
PHILIPPE BONNEAU
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Preceding Paper ◽  
Extension Theory ◽  
Quantum Double ◽  
Infinite Dimensional ◽  
Double Structure ◽  
Topological Quantum ◽  
Definition Of ◽  
Zero Class

Following a preceding paper showing how the introduction of a t.v.s. topology on quantum groups led to a remarkable unification and rigidification of the different definitions, we adapt here, in the same way, the definition of quantum double. This topological double is dualizable and reflexive (even for infinite dimensional algebras). In a simple case we show, considering the double as the "zero class" of an extension theory, the uniqueness of the double structure as a quasi-Hopf algebra. A la suite d'un précédent article montrant comment l'introduction d'une topologie d'e.v.t. sur les groupes quantiques permet une unification et une rigidification remarquables des différentes définitions, on adapte ici de la même manière la définition du double quantique. Ce double topologique est alors dualisable et reflexif (même pour des algèbres de dimension infinie). Dans un cas simple on montre, en considérant le double comme la "classe zéro" d'une théorie d'extensions, l'unicité de cette structure comme algèbre quasi-Hopf.

scholarly journals A METHOD FOR OBTAINING QUANTUM DOUBLES FROM THE YANG-BAXTER R-MATRICES

1993 ◽  
Vol 08 (14) ◽  
pp. 1315-1321 ◽  
Author(s):  
A. A. VLADIMIROV
Keyword(s):  
Hopf Algebra ◽  
Quantum Double ◽  
Constant Solution ◽  
Quasitriangular Hopf Algebra

We develop the approaches of Faddeev, Reshetikhin and Takhtajan1 and Majid2 which enable one to associate a quasitriangular Hopf algebra to every regular invertible constant solution of the quantum Yang-Baxter equations. We show that such a Hopf algebra is actually a quantum double.

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.

scholarly journals DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS

1999 ◽  
Vol 11 (05) ◽  
pp. 553-629 ◽  
Author(s):  
FRANK HAUSSER ◽  
FLORIAN NILL
Keyword(s):  
Hopf Algebra ◽  
Associative Algebra ◽  
Quantum Groups ◽  
Product Formula ◽  
Crossed Product ◽  
Group Symmetry ◽  
Algebra Structure ◽  
Quantum Double ◽  
Field Algebra ◽  
Dual Hopf Algebra

A two-sided coaction [Formula: see text] of a Hopf algebra [Formula: see text] on an associative algebra ℳ is an algebra map of the form [Formula: see text] , where (λ,ρ) is a commuting pair of left and right [Formula: see text] -coactions on ℳ, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra [Formula: see text] on ℳ by ◃ and ▹, respectively, we define the diagonal crossed product[Formula: see text] to be the algebra generated by ℳ and [Formula: see text] with relations given by [Formula: see text] We give a natural generalization of this construction to the case where [Formula: see text] is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct Δ is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on [Formula: see text] extending [Formula: see text], even though the analogue of an ordinary crossed product [Formula: see text] in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case [Formula: see text] and λ=ρ=Δ we obtain an explicit definition of the quantum double [Formula: see text] for quasi-Hopf algebras [Formula: see text] , which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that [Formula: see text] is itself a (weak) quasi-bialgebra and that any diagonal crossed product [Formula: see text] naturally admits a two-sided [Formula: see text] -coaction. In particular, the above-mentioned lattice models always admit the quantum double [Formula: see text] as a localized cosymmetry, generalizing results of Nill and Szlachányi [42]. A complete proof that [Formula: see text] is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].

Some ribbon elements for the quasi-Hopf algebra Dω(H)

2020 ◽  
pp. 2150226
Author(s):  
Daniel Bulacu ◽  
Florin Panaite
Keyword(s):  
Hopf Algebra ◽  
Quantum Double

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra [Formula: see text] defined in [D. Bulacu and F. Panaite, A generalization of the quasi-Hopf algebra [Formula: see text], Commun. Algebra 26 (1998) 4125–4141] and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for [Formula: see text].

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

scholarly journals Quantum groups and conformal field theories

1989 ◽  
Vol 329 (1605) ◽  
pp. 343-347
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Theory ◽  
Conformal Field

Rational conformal field theories can be interpreted as defining quasi-triangular Hopf algebras. The Hopf algebra is determined by the duality properties of the conformal theory.

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.

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