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On fusion categories
 Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
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Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60th birthday 1. introduction The theory of fusion categories arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi) Hopf algebras are important examples of fusion categories. Fusion categories have been studied extensively in the literature,
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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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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Cited by 75 (12 self)
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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
A categorification of quantum sl(2
 Adv. Math
"... We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs betwee ..."
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Cited by 67 (9 self)
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We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs between 1morphisms are graded lifts of a semilinear form defined on ˙U. Graded lifts of various homomorphisms and antihomomorphisms of U ˙ arise naturally in the context of our graphical calculus. For each positive integer N a representation of U˙ is constructed using iterated flag varieties that categorifies the irreducible (N + 1)dimensional representation of ˙ U.
TFT CONSTRUCTION OF RCFT CORRELATORS I: . . .
, 2002
"... We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of MooreSeiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single ..."
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Cited by 64 (19 self)
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We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of MooreSeiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are Amodules, and (generalised) defect lines are AAbimodules. The relation with threedimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in threemanifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIMrep properties. We suggest that our results can be interpreted in terms of noncommutative geometry over the modular tensor category of MooreSeiberg data.
Duality and defects in rational conformal field theory
, 2006
"... We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We sh ..."
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Cited by 61 (18 self)
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We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and orderdisorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetracritical Ising model and the critical threestates
Weakly grouptheoretical and solvable fusion categories
"... To Izrail Moiseevich Gelfand on his 95th birthday with admiration ..."
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To Izrail Moiseevich Gelfand on his 95th birthday with admiration