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143
THE COLORED JONES POLYNOMIALS AND THE SIMPLICIAL VOLUME OF A Knot
, 1999
"... We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Theref ..."
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Cited by 101 (10 self)
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We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Therefore Kashaev’s conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can prove that a knot is trivial if and only if all of its Vassiliev invariants are trivial.
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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Cited by 76 (17 self)
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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,
Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 62 (10 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
An invariant of integral homology 3spheres which is universal for all finite type invariants, preprint
, 1996
"... Abstract. In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite ..."
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Cited by 54 (4 self)
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Abstract. In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite type invariants, i.e. Ωn is an invariant order 3n which dominates all other invariants of the same order. Some corollaries are discussed. 1.
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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Cited by 27 (2 self)
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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.
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
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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Cited by 20 (4 self)
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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.
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 19 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
Chord Diagram Invariants of Tangles and Graphs
 Duke Math. J
, 1998
"... this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we fix a commutative ring R and consider categories T (R) and A(R) whose morphisms are f ..."
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Cited by 18 (3 self)
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this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we fix a commutative ring R and consider categories T (R) and A(R) whose morphisms are formal linear combinations of framed oriented tangles and chord diagrams with coefficients in R, cf. Section 2. The set of morphisms in T (R) has a canonical filtration given by the powers of an ideal I which we call the augmentation ideal. Functions on morphisms in T (R) vanishing on I
Perverse sheaves on affine flags and Langlands dual group
"... Abstract. This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a (split) simple group in terms the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal com ..."
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Cited by 17 (10 self)
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Abstract. This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a (split) simple group in terms the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra H; of the antispherical module for H; and of the space of Iwahoriinvariant Whittaker functions. As a byproduct we obtain some new properties of central sheaves introduced in [G]. Acknowledgements. This project was conceived during the IAS special year in Representation Theory (1998/99) led by G. Lusztig, as a result of conversations with D. Gaitsgory, M. Finkelberg and I. Mirkovic. The outcome was strongly influenced by conversations with A. Beilinson and V. Drinfeld. The stimulating interest of A. Braverman, D. Kazhdan, G. Lusztig and V. Ostrik was crucial for keeping the project alive. We are very grateful to all these people. We thank I. Mirkovic and D. Gaitsgory for the permission to use their unpublished results; and M. Finkelberg and D. Gaitsgory for taking the trouble to read the text and point out various lapses in the exposition. The second author was supported by NSF and Clay Institute. 1.