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35
Galois theory for braided tensor categories and the modular closure
 Adv. Math
, 2000
"... Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC ..."
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Cited by 57 (9 self)
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Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC with positive ∗operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no nontrivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
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 43 (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
Representation Theory of ChernSimons Observables
, 1995
"... In [2], [3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian ChernSimons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2dimensional surfac ..."
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Cited by 39 (0 self)
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In [2], [3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian ChernSimons theory. This algebra provides the quantization of the algebra of functions on the moduli space of flat connections on a 2dimensional surface. In this paper we classify unitary representations of this new algebra and identify the corresponding representation spaces with the spaces of conformal blocks of the WZW model. The mapping class group of the surface is proved to act on the moduli algebra by inner automorphisms. The generators of these automorphisms are unitary elements of the moduli algebra. They are constructed explicitly and proved to satisfy the relations of the (unique) central extension of the mapping class group.
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 36 (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
On nonsemisimple fusion rules and tensor categories
, 2006
"... Category theoretic aspects of nonrational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecom ..."
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Cited by 30 (0 self)
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Category theoretic aspects of nonrational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be blockdiagonalized. A conjectural connection between the blockdiagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models.
Factorizable ribbon quantum groups in logarithmic conformal field theories
 THEOR. MATH. PHYS
, 2007
"... 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 ..."
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Cited by 22 (12 self)
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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 logarit
hmic conformal field models.
On the TQFT representations of the mapping class groups
 Pacific J. Math
"... We prove that the image of the mapping class group by the representations arising in the SU(2)TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r ̸ = 2,3,4,6 (and r ̸ = 10 for g = 2). In particular it follows that the quotient groups Mg/N (tr) by the normalizer of the rth ..."
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Cited by 21 (7 self)
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We prove that the image of the mapping class group by the representations arising in the SU(2)TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r ̸ = 2,3,4,6 (and r ̸ = 10 for g = 2). In particular it follows that the quotient groups Mg/N (tr) by the normalizer of the rth power of a Dehn twist t are infinite if g ≥ 3 and r ̸ = 2,3,4,6,8,12. 1. Introduction. Witten [50] constructed a TQFT in dimension 3 using path integrals and afterwards several rigorous constructions arose, like those using the quantum group approach ([39, 25]), the TemperleyLieb algebra ([30, 31]), the theory based on the Kauffman bracket ([4, 5]) or that obtained from the mapping
Selfdual Modules of Semisimple Hopf Algebras
, 2001
"... We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial selfdual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf ..."
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Cited by 18 (2 self)
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We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial selfdual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension. 1 Suppose that H is a finitedimensional Hopf algebra that is defined over the field K. We denote its comultiplication by ∆, its counit by ε, and its antipode by S. For the comultiplication, we use the sigma notation of R. G. Heyneman and M. E. Sweedler in the following variant: ∆(h) = h (1) ⊗ h (2) We view the dual space H ∗ as a Hopf algebra whose unit is the counit of H, whose counit is the evaluation at 1, whose antipode is the transpose of the antipode of H, and whose multiplication and comultiplication are determined
KazhdanLusztigdual quantum group for logarithmic . . .
, 2006
"... We derive and study a quantum group g p,q that is Kazhdan–Lusztigdual to the Walgebra Wp,q of the logarithmic (p, q) conformal field theory model. The number (2pq) of irreducible representations of g p,q is the same as the number of irreducible Wp,qrepresentations on which the maximum vertexope ..."
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Cited by 17 (1 self)
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We derive and study a quantum group g p,q that is Kazhdan–Lusztigdual to the Walgebra Wp,q of the logarithmic (p, q) conformal field theory model. The number (2pq) of irreducible representations of g p,q is the same as the number of irreducible Wp,qrepresentations on which the maximum vertexoperator ideal acts nontrivially. We find the center of g p,q and show that the modular group representation on it is equivalent to the modular group representation on the Wp,q characters and “pseudocharacters. ” The factorization of the g p,q ribbon element leads to a factorization of the modular group representation on the center. We also find the g p,q Grothendieck ring, which is presumably