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Finite tensor categories
 Moscow Math. Journal
"... These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We wil ..."
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These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We will also assume the basics of the theory of abelian categories (for a more detailed treatment see the book [F]). If C is a category, the notation X ∈ C will mean that X is an object of C, and the set of morphisms between X, Y ∈ C will be denoted by Hom(X, Y). Throughout the notes, for simplicity we will assume that the ground field k is algebraically closed unless otherwise specified, even though in many cases this assumption will not be needed. 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of thinking
Module categories over representations of SLq(2) and graphs
"... Abstract. We classify semisimple module categories over the tensor category of representations of quantum SL(2). 1. ..."
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Abstract. We classify semisimple module categories over the tensor category of representations of quantum SL(2). 1.
ON THE CLASSIFICATION OF GALOIS OBJECTS OVER THE QUANTUM GROUP OF A NONDEGENERATE BILINEAR FORM
, 2006
"... Abstract. We study Galois and biGalois objects over the quantum group of a nondegenerate bilinear form, including the quantum groupOq(SL(2)). We obtain the classification of these objects up to isomorphism and some partial results for their classification up to homotopy. ..."
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Abstract. We study Galois and biGalois objects over the quantum group of a nondegenerate bilinear form, including the quantum groupOq(SL(2)). We obtain the classification of these objects up to isomorphism and some partial results for their classification up to homotopy.
Hopf algebra deformations of binary polyhedral groups
"... Abstract. We show that semisimple Hopf algebras having a selfdual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and descr ..."
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Abstract. We show that semisimple Hopf algebras having a selfdual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras. 1. Introduction and
Cosemisimple Hopf Algebras with Antipode of Arbitrary Finite Order
"... Abstract. Let m ≥ 1 be a positive integer. We show that, over an algebraically closed field of characteristic zero, there exist cosemisimple Hopf algebras having an antipode of order 2m. We also discuss the Schur indicator for such Hopf algebras. ..."
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Abstract. Let m ≥ 1 be a positive integer. We show that, over an algebraically closed field of characteristic zero, there exist cosemisimple Hopf algebras having an antipode of order 2m. We also discuss the Schur indicator for such Hopf algebras.
1.36. Twists for bialgebras and Hopf algebras 71 1.37. Quantum traces 72 1.38. Pivotal categories and dimensions 73 1.39. Spherical categories 74
"... 1.42. Grothendieck rings of semisimple tensor categories 76 1.43. Semisimplicity of multifusion rings 79 1.44. The FrobeniusPerron theorem 80 1.45. Tensor categories with finitely many simple objects. FrobeniusPerron dimensions 82 1.46. Deligne’s tensor product of finite abelian categories 86 1.47 ..."
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1.42. Grothendieck rings of semisimple tensor categories 76 1.43. Semisimplicity of multifusion rings 79 1.44. The FrobeniusPerron theorem 80 1.45. Tensor categories with finitely many simple objects. FrobeniusPerron dimensions 82 1.46. Deligne’s tensor product of finite abelian categories 86 1.47. Finite (multi)tensor categories 87 1.48. Integral tensor categories 89
unknown title
"... Exercise 2.11.1. Show that for any M ∈ M the object Hom(M, M) with the multiplication defined above is an algebra (in particular, define the unit morphism!). Theorem 2.11.2. Let M be a module category over C, and assume that M ∈ M satisfies two conditions: 1. The functor Hom(M, •) is right exact (no ..."
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Exercise 2.11.1. Show that for any M ∈ M the object Hom(M, M) with the multiplication defined above is an algebra (in particular, define the unit morphism!). Theorem 2.11.2. Let M be a module category over C, and assume that M ∈ M satisfies two conditions: 1. The functor Hom(M, •) is right exact (note that it is automatically left exact). 2. For any N ∈ M there exists X ∈ C and a surjection X ⊗M → N. Let A = Hom(M, M). Then the functor F: = Hom(M, •) : M → ModC (A) is an equivalence of module categories. Proof. We will proceed in steps: (1) The map F: Hom(N1, N2) → HomA(F (N1), F (N2)) is an isomorphism for any N2 ∈ M and N1 of the form X ⊗ M, X ∈ C. Indeed, F (N1) = Hom(M, X ⊗ M) = X ⊗ A and the statement follows from the calculation: HomA(F (N1), F (N2)) = HomA(X ⊗ A, F (N2)) = Hom(X, F (N2)) = = Hom(X, Hom(M, N2)) = Hom(X ⊗ M, N2) = Hom(N1, N2). (2) The map F: Hom(N1, N2) → HomA(F (N1), F (N2)) is an isomorphism for any N1, N2 ∈ M. By condition 2, there exist objects X, Y ∈ C and an exact sequence Since F is exact, the sequence
ON THE REPRESENTATION CATEGORIES OF MATRIX QUANTUM GROUPS OF TYPE A
, 2005
"... Abstract. A quantum groups of type A is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the birank of the Hecke symmetry. 1. ..."
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Abstract. A quantum groups of type A is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the birank of the Hecke symmetry. 1.