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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 26 (8 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
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
On tensor categories attached to cells in affine Weyl groups
"... Abstract. This note is devoted to Lusztig’s bijection between unipotent conjugacy classes in a simple complex algebraic group and 2sided cells in the affine Weyl group of the Langlands dual group; and also to the description of the reductive quotient of the centralizer of the unipotent element in t ..."
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Abstract. This note is devoted to Lusztig’s bijection between unipotent conjugacy classes in a simple complex algebraic group and 2sided cells in the affine Weyl group of the Langlands dual group; and also to the description of the reductive quotient of the centralizer of the unipotent element in terms of convolution of perverse sheaves on affine flag variety of the dual group conjectured by Lusztig in [L4]. Our main tool is a recent construction by Beilinson, Gaitsgory and Kottwitz, the socalled sheaftheoretic construction of the center of an affine Hecke algebra (see [Ga]). We show how this remarkable construction provides a geometric interpretation of the bijection, and allows to prove the conjecture. Acknowledgement. I am much indebted to Michael Finkelberg; among the many things he taught me are the theory of cells in Weyl groups and Lusztig’s conjecture (partially proved below). This note owes a lot to Dennis Gaitsgory, who explained to me the “sheaftheoretic center ” construction, and made some helpful suggestions. I also thank Alexander Beilinson and Vladimir Drinfeld for useful comments, and Viktor Ostrik for stimulating interest. The results of this note were obtained (in a preliminary form) during the author’s participation in the special year on Geometric Methods in Representation Theory (98/99) at IAS; I thank IAS for its hospitality, and NSF grant for financial support.
Fusion categories of rank 2
, 2003
"... Abstract. We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P. Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is fin ..."
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Abstract. We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P. Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite. 1.
Weakly grouptheoretical and solvable fusion categories
"... To Izrail Moiseevich Gelfand on his 95th birthday with admiration ..."
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Cited by 17 (4 self)
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To Izrail Moiseevich Gelfand on his 95th birthday with admiration
On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
Grouptheoretical properties of nilpotent modular categories, eprint arXiv:0704.0195v2 [math.QA
"... Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects o ..."
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Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects ofC have integral FrobeniusPerron dimensions then C is grouptheoretical in the sense of [ENO]. As a consequence, we obtain that semisimple quasiHopf algebras of prime power dimension are grouptheoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasiLie bialgebras in terms of Manin pairs given in [Dr]). 1. introduction In this paper we work over an algebraically closed field k of characteristic 0. By a fusion category we mean a klinear semisimple rigid tensor category C with finitely many isomorphism classes of simple objects, finite dimensional spaces of morphisms, and such that the unit object 1 of C is simple. We refer the reader to [ENO] for a general theory of such categories. A fusion category is pointed if all its simple objects are invertible. A pointed fusion category is equivalent to Vec ω G, i.e., the category of Ggraded vector spaces with the associativity constraint given by some cocycle ω ∈ Z 3 (G, k × ) (here G is a finite group). 1.1. Main results. Theorem 1.1. Any braided nilpotent fusion category has a unique decomposition into a tensor product of braided fusion categories whose FrobeniusPerron dimensions are powers of distinct primes. The notion of nilpotent fusion category was introduced in [GN]; we recall it in Subsection 2.2. Let us mention that the representation category Rep(G) of a finite group G is nilpotent if and only if G is nilpotent. It is also known that fusion categories of prime power FrobeniusPerron dimension are nilpotent [ENO]. On the other hand, Vec ω G is nilpotent for any G and ω. Therefore it is not true that any nilpotent fusion category is a tensor product of fusion categories of prime power dimensions.
Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
 Comm. Math. Phys
"... Abstract. We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω (G), where G is a finite group and ω is a 3cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of ..."
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Abstract. We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω (G), where G is a finite group and ω is a 3cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between
On Vafa’s theorem for tensor categories
 MR MR1906068 (2003i:18009) SIUHUNG NG AND PETER SCHAUENBURG
"... In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), ..."
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In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple),