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31
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
"... ..."
Braided Hopf algebras over non abelian finite groups
 Acad. Nac. Ciencias (Córdoba
"... Abstract. In the last years a new theory of Hopf algebras has begun to be developed: that of Hopf algebras in braided categories, or, briefly, braided Hopf algebras. This is a survey of general aspects of the theory with emphasis in H HYD, the Yetter–Drinfeld category over H, where H is the group al ..."
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Cited by 40 (11 self)
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Abstract. In the last years a new theory of Hopf algebras has begun to be developed: that of Hopf algebras in braided categories, or, briefly, braided Hopf algebras. This is a survey of general aspects of the theory with emphasis in H HYD, the Yetter–Drinfeld category over H, where H is the group algebra of a non abelian finite group Γ. We discuss a special class of braided graded Hopf algebras from different points of view following Lusztig, Nichols and Schauenburg. We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider. Sinopsis. En los últimos años comenzó a ser desarrollada una nueva teoría de álgebras de Hopf en categorías trenzadas, o brevemente, álgebras de Hopf trenzadas. Presentamos aquí aspectos generales de la teoría con énfasis en H HYD, la categoría de Yetter–Drinfeld sobre H, donde H es el álgebra de grupo de un grupo finito no abeliano Γ. Discutimos una clase especial de álgebras de Hopf trenzadas graduadas desde diferentes puntos de vista, siguiendo a Lusztig, Nichols y Schauenburg. Presentamos algunos ejemplos de dimensión finita que aparecen en un trabajo inédito de Milinski y Schneider. 0. Introduction and notations 0.1. Introduction.
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.
On Inner Product In Modular Tensor Categories. II Inner Product On Conformal Blocks.
 I & II, math.QA/9508017 and qalg/9611008
, 1995
"... this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and la ..."
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Cited by 29 (0 self)
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this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and later refined by Kazhdan and Lusztig ([KL14]) and Finkelberg ([F]) in Section 9. In particular, spaces of homomorphisms in this category are the spaces of conformal blocks of WZW model. Thus, the general theory developed in Section 2 of [K] gives us an inner product on the space of conformal blocks, and so defined inner product is modular invariant. This definition is constructive: we show how it can be rewritten so that it only involves Drinfeld associator, or, equivalently, asymptotics of solutions of KnizhnikZamolodchikov equations. Since there are integral formulas for the solutions of KZ equations, this shows that the inner product on the space of conformal blocks can be written explicitly in terms of certain integrals. In the case g = sl 2 these integrals can be calculated (see [V]), using Selberg integral, and the answer is written in terms of \Gammafunctions. Thus, in this case we can write explicit formulas for inner product on the space of conformal blocks. These expressions are 1991 Mathematics Subject Classification. Primary 81R50, 05E35, 18D10; Secondary 57M99
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
Integrals for braided Hopf algebras
 J. Pure Appl. Algebra
, 2000
"... Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided ve ..."
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Cited by 17 (3 self)
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Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integration with crossproduct and transmutation are studied. 1991 Mathematics Subject Classification. Primary 16W30, 18D15, 17B37; Secondary 18D35.
Hopf (Bi)Modules and Crossed Modules in Braided Monoidal Categories
 J. Pure Appl. Algebra
, 1995
"... Hopf (bi)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra ..."
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Cited by 16 (1 self)
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Hopf (bi)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. A generalization of the MajidRadford criterion for a braided Hopf algebra to be a cross product is obtained as an application of these results. Keywords: Braided category, Braided Hopf algebra, Crossed Module, Hopf (Bi)Module Mathematical Subject Classification (1991): 16W30, 17B37, 18D10, 81R50 1 Introduction For bialgebras over a field k the smash product and the smash coproduct are investigated extensively in the literature [Rad, Mol]. Let H be a bialgebra, B be an Hright module algebra and an Hright comodule coalgebra. If the smash product algebra structure and the smash coproduct coalgebra structure on H\Omega B form a bialgebra then Ra...
A treatise on quantum Clifford Algebras
"... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very e ..."
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Cited by 13 (10 self)
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on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very efficient and may be used in Robotics, left and right contractions, left and right cocontractions, Clifford and coClifford products, etc. The Chevalley deformation, using a Clifford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a biconvolution. Antipode and crossing are consequences of the product and coproduct structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the definition of nonlocal products and interacting Hopf gebras which are generically nonperturbative. A ‘spinorial ’ generalization of the antipode is given. The nonexistence of nontrivial integrals in lowdimensional Clifford cogebras is shown. Generalized cliffordization is discussed which is based on nonexponentially generated bilinear forms in general resulting in non unital, nonassociative products. Reasonable assumptions lead to bilinear forms based on 2cocycles. Cliffordization is used to derive time and normalordered generating functionals for the SchwingerDyson hierarchies of nonlinear spinor field theory and spinor electrodynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory. MSC2000: 16W30 Coalgebras, bialgebras, Hopf algebras; 1502 Research exposition (monographs, survey articles);
Noncommutative differentials and YangMills on permutation groups SN, math/0105253. 11
 Research Institute for Mathematical Sciences Kyoto University Sakyoku, Kyoto
, 1989
"... Abstract We study noncommutative differential structures on permutation groups SN, defined by conjugacy classes. The 2cycles class defines an exterior algebra which is a super analogue of the quadratic algebra of Fomin et al. in theory of the cohomology of the flag variety. Noncommutative de Rahm c ..."
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Cited by 13 (3 self)
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Abstract We study noncommutative differential structures on permutation groups SN, defined by conjugacy classes. The 2cycles class defines an exterior algebra which is a super analogue of the quadratic algebra of Fomin et al. in theory of the cohomology of the flag variety. Noncommutative de Rahm cohomology and moduli of flat connections are computed for N < 6. For any finite group, the flat connections of submaximal cardinality form a natural representation associated to each conjugacy class, sometimes irreducible. 1