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58
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 ..."
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Cited by 87 (13 self)
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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
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QUANTUM AND BRAIDED LIE ALGEBRAS
, 1993
"... We introduce the notion of a braided Lie algebra consisting of a finitedimensional vector space L equipped with a bracket [ , ] : L ⊗ L → L and a YangBaxter operator Ψ: L ⊗ L → L ⊗ L obeying some axioms. We show that such an object has an enveloping braidedbialgebra U(L). We show that every gener ..."
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Cited by 52 (29 self)
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We introduce the notion of a braided Lie algebra consisting of a finitedimensional vector space L equipped with a bracket [ , ] : L ⊗ L → L and a YangBaxter operator Ψ: L ⊗ L → L ⊗ L obeying some axioms. We show that such an object has an enveloping braidedbialgebra U(L). We show that every generic Rmatrix leads to such a braided Lie algebra with [ , ] given by structure constants c IJ K determined from R. In this case U(L) = B(R) the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural rightregular action of a braidedLie algebra L by braided vector fields, the braidedKilling form and the quadratic Casimir associated to L. These constructions recover the relevant notions for usual, colour and superLie algebras as special cases. In addition, the standard quantum deformations Uq(g) are understood as the enveloping algebras of such underlying braided Lie algebras with [ ,]
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 ..."
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Cited by 35 (25 self)
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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.
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 ..."
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Cited by 28 (21 self)
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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
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Classification of Bicovariant DIFFERENTIAL CALCULI
, 1996
"... We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 11 correspondence with irreducible representations V of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension dimV 2. The differential calculi ..."
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Cited by 25 (19 self)
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We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 11 correspondence with irreducible representations V of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension dimV 2. The differential calculi on a finite group algebra CG are also classified and shown to be in correspondence with pairs consisting of an irreducible representation V and a continuous parameter in CP dim V −1. They have dimension dimV. For a classical Lie group we obtain an infinite family of nonstandard calculi. General constructions for bicovariant calculi and their quantum tangent spaces are also obtained.
On twisting solutions to the YangBaxter equation
"... Sufficient conditions for an invertible twotensor F to relate two solutions to the YangBaxter equation via the transformation R → F −1 21 RF are formulated. Those conditions include relations arising from twisting of certain quasitriangular bialgebras. 1 The twist procedure for (quasi)Hopf algebr ..."
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Cited by 24 (1 self)
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Sufficient conditions for an invertible twotensor F to relate two solutions to the YangBaxter equation via the transformation R → F −1 21 RF are formulated. Those conditions include relations arising from twisting of certain quasitriangular bialgebras. 1 The twist procedure for (quasi)Hopf algebras developed by Drinfeld [1, 2] (see also [3, 4]) allows to deform the coproduct via a similarity transformation with the multiplication unchanged. Twist finds its applications in solvable models and noncommutative geometry because it appears to be very friendly to all the algebraic properties
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 ..."
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Cited by 22 (1 self)
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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
DrinfeldSokolov reduction for difference operators and deformations of W–algebras I. The case of Virasoro algebra
"... Abstract. We propose a q–difference version of the DrinfeldSokolov reduction scheme, which gives us q–deformations of the classical W–algebras by reduction from PoissonLie loop groups. We consider in detail the case of SL2. The nontrivial consistency conditions fix the choice of the classical rma ..."
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Cited by 22 (4 self)
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Abstract. We propose a q–difference version of the DrinfeldSokolov reduction scheme, which gives us q–deformations of the classical W–algebras by reduction from PoissonLie loop groups. We consider in detail the case of SL2. The nontrivial consistency conditions fix the choice of the classical rmatrix defining the PoissonLie structure on the loop group LSL2, and this leads to a new elliptic classical r– matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra. 1.