Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.
No abstract
To any finite group Γ ⊂ Sp(V ) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V ]#Γ, smash product of Γ with the polynomial algebra on V . The parameter κ runs over points of P r , where r = number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V /Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space h and V = h ⊕ h * , then the algebras Hκ are certain 'rational' degenerations of the double affine Hecke algebra introduced earlier by Cherednik.Let Γ = Sn, the Weyl group of g = gl n . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from D(g) g , the algebra of invariant polynomial differential operators on gl n , to the algebra of Sn-invariant differential operators with rational coefficients on the space C n of diagonal matrices. The second order Laplacian on g goes, under the deformed homomorphism, to the Calogero-Moser differential operator on C n , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: D(g) g ։ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ = Sn. This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of 'quantum' Hamiltonian reduction.In the 'classical' limit κ → ∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the CalogeroMoser space, whose fibers carry the regular representation of Sn. Moreover, we prove that the algebra H∞ is isomorphic to the endomorphism algebra of that vector bundle.
0. Introduction. The quantum Yang-Baxter equation (QYBE) is one of the basic equations in mathematical physics that lies in the foundation of the theory of quantum groups. This equation involves a linear operator R : V ⊗ V → V ⊗ V , where V is a vector space, and has the form
Abstract.We apply the yoga of classical homotopy theory to classification problems of Gextensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic.C / of invertible C -bimodule categories, called the Brauer-Picard groupoid of C , such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic.C /. This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically.One of the central results of the article is that the 2-truncation of BrPic.C / is canonically equivalent to the 2-groupoid of braided auto-equivalences of the Drinfeld center Z.C/ of C. In particular, this implies that the Brauer-Picard group BrPic.C / (i.e., the group of equivalence classes of invertible C -bimodule categories) is naturally isomorphic to the group of braided auto-equivalences of Z.C /. Thus, if C D Vec A , where A is a finite abelian group, then BrPic.C / is the orthogonal group O.A˚A /. This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case G D Z 2 , we re-derive (without computations) the classical result of Tambara andYamagami. Moreover, we explicitly describe the category of all .Vec A 1 ; Vec A 2 /-bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.Mathematics Subject Classification (2010). 18D10, 55S35.
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