Results 1  10
of
83
Algebras and Hopf algebras IN BRAIDED CATEGORIES
, 1995
"... This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras i ..."
Abstract

Cited by 87 (13 self)
 Add to MetaCart
This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied, the notion of ‘braidedcommutative ’ or ‘braidedcocommutative ’ Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group Aut (C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.
BRAIDED MATRIX STRUCTURE OF THE SKLYANIN ALGEBRA AND OF THE QUANTUM LORENTZ GROUP
, 1992
"... Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of supergroups and supermatrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups Uq(g). They h ..."
Abstract

Cited by 35 (25 self)
 Add to MetaCart
Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of supergroups and supermatrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups Uq(g). They have the same FRT generators l ± but a matrix braidedcoproduct ∆L = L⊗L where L = l + Sl −, and are selfdual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices BMq(2); it is a braidedcommutative bialgebra in a braided category. As a second application, we show that the quantum double D(Uq(sl2)) (also known as the ‘quantum Lorentz group’) is the semidirect product as an algebra of two copies of Uq(sl2), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semiclassical limits as doubles of the Lie algebras of Poisson Lie groups.
Quantum geometry of algebra factorisations and coalgebra bundles
 Commun. Math. Phys
, 2000
"... We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and el ..."
Abstract

Cited by 34 (15 self)
 Add to MetaCart
We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and elements of Riemannian geometry. As an example, we construct qmonopoles on all the Podle´s quantum spheres S 2 q,s. 1.
Extensions of locally compact quantum groups and the bicrossed product construction
, 2001
"... ..."
The Quantum Double as Quantum Mechanics
"... We introduce ∗structures on braided groups and braided matrices. Using this, we show that the quantum double D(Uq(su2)) can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in qMinkowski space (a threesphere in the Lorentz metric), and with the role of ..."
Abstract

Cited by 28 (21 self)
 Add to MetaCart
We introduce ∗structures on braided groups and braided matrices. Using this, we show that the quantum double D(Uq(su2)) can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in qMinkowski space (a threesphere in the Lorentz metric), and with the role of angular momentum played by Uq(su2). This provides a new example of a quantum system whose algebra of observables is a Hopf algebra. Furthermore, its dual Hopf algebra can also be viewed as a quantum algebra of observables, of another quantum system. This time the position space is a qdeformation of SL(2, R) and the momentum group is Uq(su ∗ 2) where su ∗ 2 is the Drinfeld dual Lie algebra of su2. Similar results hold for the quantum double and its dual of a general quantum group.
QED Hopf algebras on planar binary trees
 Preprint 2001/15 of Institut Girard Desargues, arXiv:math.QA/0112043
"... In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semidirect coproduct of Hopf algebras. 1 ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semidirect coproduct of Hopf algebras. 1
Classification of finitedimensional triangular Hopf algebras with the Chevalley property
 Mathematical Research Letters
"... Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose ..."
Abstract

Cited by 18 (9 self)
 Add to MetaCart
Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne’s theorem on Tannakian categories and on Movshev’s theory in an essential way. We explain Movshev’s theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property. 1.
SOLUTIONS OF THE YANGBAXTER EQUATIONS FROM BRAIDEDLIE ALGEBRAS AND BRAIDED GROUPS
, 1993
"... We obtain an Rmatrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)Lie algebra or braidedLie algebra. The same result applies for every (super)Hopf algebra or braidedHopf algebra. We recover some known representations such as th ..."
Abstract

Cited by 16 (10 self)
 Add to MetaCart
We obtain an Rmatrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)Lie algebra or braidedLie algebra. The same result applies for every (super)Hopf algebra or braidedHopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a nontrivial one on the ring k[x] of polynomials in one variable, regarded as a braidedline. Representations of the extended Artin braid group for braids in the complement of S 1 are also obtained by the same method.
A survey on multiplier Hopf algebras
 Proceedings of the conference in Brussels on Hopf algebras, Hopf Algebras and Quantum groups, eds. Caenepeel/Van Oystaeyen (2000
"... In this paper, we generalize Majid’s bicrossproduct construction. We start with a pair (A, B) of two regular multiplier Hopf algebras. We assume that B is a right Amodule algebra and that A is a left Bcomodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
In this paper, we generalize Majid’s bicrossproduct construction. We start with a pair (A, B) of two regular multiplier Hopf algebras. We assume that B is a right Amodule algebra and that A is a left Bcomodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A#B. The left coaction of B on A gives a possible coproduct ∆ # on A#B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for ∆ # to be a proper coproduct on A#B. The result is again a regular multiplier Hopf algebra. Majid’s construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair (C, D) of regular multiplier Hopf algebras where now C is a left Dmodule algebra while D is a right Ccomodule coalgebra. We will show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D will yield a duality between A#B and the smash product C#D. We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The ∗algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature. Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G = KH and H ∩ K = {e} (where e is the identity of G) will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention.
On Group Theoretical Hopf Algebras and Exact Factorizations of Finite Groups
 J. of Algebra
, 2003
"... Abstract. We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the DijkgraafPasquierRoche quasiHopf algebra D ω (Σ), for some finite group Σ and some ω ∈ Z 3 (Σ, k ×). We show that semisimple Hopf algebras obtained as bicrossed products ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Abstract. We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the DijkgraafPasquierRoche quasiHopf algebra D ω (Σ), for some finite group Σ and some ω ∈ Z 3 (Σ, k ×). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as a twisting of D ω (Σ), for an appropriate 3cocycle ω coming from the Kac exact sequence. 1.