Results 1  10
of
29
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.
BEYOND SUPERSYMMETRY AND QUANTUM SYMMETRY (AN INTRODUCTION TO BRAIDEDGROUPS AND BRAIDEDMATRICES)
, 1993
"... ..."
Mapping class group action on quantum doubles
 Commun. Math. Phys
, 1995
"... Abstract: We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly nonsemisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT’ ..."
Abstract

Cited by 29 (2 self)
 Add to MetaCart
Abstract: We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly nonsemisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT’s is circumvented. We find compact formulae for the S ±1matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford’s relations between the canonical forms and the moduli of integrals. We analyze the projective SL(2,Z)action on the center of Uq(sl2) for q an l = 2m + 1st root of unity. It appears that the 3m + 1dimensional representation decomposes into an m + 1dimensional finite representation and a 2mdimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of SL(2,Z) and the finite, mdimensional representation, obtained from the truncated TQFT Since the seminal paper of Atiyah [A] on the abstract definition of a topological quantum field theory (TQFT) much progress has been made in finding non trivial examples and
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.
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
"... ..."
Invariants of 3–manifolds and projective representations of mapping class groups via quantum groups at roots of unity
 Comm. Math. Phys
, 1995
"... Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphis ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphisms in the Hmodule H ∗⊗g, if H ∗ is endowed with the coadjoint Hmodule structure. There exists a projective representation of the mapping class group Mg,n of a surface of genus g with n holes labelled by finite dimensional Hmodules X1,..., Xn in the vector space HomH(X1 ⊗ · · · ⊗ Xn, H ∗⊗g). An invariant of closed oriented 3manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most of uq(g) at roots of unity q of even degree) are described. After works of Moore and Seiberg [44], Witten [62], Reshetikhin and Turaev [51], Walker [61], Kohno [22, 23] and Turaev [59] it became clear that any semisimple abelian ribbon category with finite number of simple objects satisfying some nondegeneracy condition gives rise to projective representations of mapping class groups
KAZHDAN–LUSZTIG CORRESPONDENCE FOR THE REPRESENTATION CATEGORY OF THE TRIPLET WALGEBRA IN LOGARITHMIC CFT
, 2006
"... ... is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finitedimensional representations of the restricted quantum group Uqsℓ(2) at q = e iπ p. We fully describe the category Cp by classifying all indecomposable representations. These ar ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
... is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finitedimensional representations of the restricted quantum group Uqsℓ(2) at q = e iπ p. We fully describe the category Cp by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the W(p) and Uqsℓ(2)representation categories is conjectured for all p �2 and proved for p = 2, the implications including the identifications of the quantumgroup center with the logarithmic conformal field theory center and of the universal Rmatrix with the braiding matrix.
Factorizable ribbon quantum groups in logarithmic conformal field theories
 THEOR. MATH. PHYS
, 207
"... We review the properties of quantum groups occurring as Kazhdan–Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
We review the properties of quantum groups occurring as Kazhdan–Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.
On the center of the small quantum group
 J. of Algebra
"... Abstract. Using the quantum Fourier transform F [LM], we describe the block decomposition and multiplicative structure of a subalgebra ˜ Z+ F ( ˜ Z) of the center of the small quantum group U fin q (g) at a root of unity. It contains the known subalgebra ˜ Z [BG], which is isomorphic to the algebra ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Abstract. Using the quantum Fourier transform F [LM], we describe the block decomposition and multiplicative structure of a subalgebra ˜ Z+ F ( ˜ Z) of the center of the small quantum group U fin q (g) at a root of unity. It contains the known subalgebra ˜ Z [BG], which is isomorphic to the algebra of characters of finite dimensional U fin q (g)modules. We prove that the intersection ˜ Z ∩ F ( ˜ Z) coincides with the annihilator of the radical of ˜ Z. Applying representationtheoretical methods, we show that ˜ Z surjects onto the algebra of endomorphisms of certain indecomposable projective modules over U fin q (g). In particular this leads to the conclusion that the center of U fin q (g) coincides with ˜ Z+ F ( ˜ Z) in the case g = sl2.