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From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Skein construction of idempotents in BirmanMurakamiWenzl algebras
 MR MR1866492 (2002h:57018) FREDERICK M. GOODMAN AND HOLLY HAUSCHILD MOSLEY
"... Abstract. We give skein theoretic formulas for minimal idempotents in the BirmanMurakamiWenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula f ..."
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Cited by 12 (2 self)
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Abstract. We give skein theoretic formulas for minimal idempotents in the BirmanMurakamiWenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas.
From Quantum Groups to Unitary Modular Tensor Categories
 CONTEMPORARY MATHEMATICS
"... Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently propos ..."
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Cited by 8 (6 self)
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Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are “physically feasible.” We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.
On a family of nonunitarizable ribbon categories
 Math Z
, 2005
"... Abstract. We consider several families of categories. The first are quotients of H. Andersen’s tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMWalgebras. Our main result is to show that ..."
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Cited by 7 (5 self)
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Abstract. We consider several families of categories. The first are quotients of H. Andersen’s tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMWalgebras. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. By appealing to similar results of Blanchet and Beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of Lie type C at odd roots of unity. The morphism spaces in these categories can be equipped with a Hermitian form, and we are able to show that these categories are never unitary, and no braided tensor category sharing the Grothendieck semiring common to these families is unitarizable. 1.
Knot polynomial identities and quantum group coincidences
"... Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between smal ..."
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Cited by 4 (2 self)
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Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D2n planar algebras. We discuss the origins of these coincidences, explaining the role of SO levelrank duality, KirbyMelvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves G2 and does not appear to be related to levelrank duality. AMS Classification 18D10; 57M27 17B10 81R05 57R56
Almost integral TQFTs from simple Lie algebras. Submitted for publication. Corrado De Concini and Victor G. Kac. Representations of quantum groups at roots of 1
 In Operator algebras, unitary representations, enveloping algebras, and invariant theory
, 1989
"... Abstract. Almost integral TQFT was introduced by Gilmer [G]. For each simple Lie algebra g and some prime integer we associate an almost integral TQFT which derives the projective WittenReshetikhinTuraev invariant τ Pg for closed 3manifolds. As a corollary, one can show that τ Pg is an algebraic ..."
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Cited by 3 (2 self)
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Abstract. Almost integral TQFT was introduced by Gilmer [G]. For each simple Lie algebra g and some prime integer we associate an almost integral TQFT which derives the projective WittenReshetikhinTuraev invariant τ Pg for closed 3manifolds. As a corollary, one can show that τ Pg is an algebraic integer for certain prime integers. The result in satisfies some Murasugi type equivalence relation if M is a homology sphere and admits a cyclic group action with fixed point set a circle. this paper can be used to prove that τ Pg M 1.
QUOTIENTS OF REPRESENTATION RINGS
"... Abstract. We give a proof using socalled fusion rings and qdeformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(∞)). This is obtained here as a limiting case for analogous quotient maps for fusion categories, ..."
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Cited by 1 (1 self)
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Abstract. We give a proof using socalled fusion rings and qdeformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(∞)). This is obtained here as a limiting case for analogous quotient maps for fusion categories, with the level going to ∞. This in turn allows a detailed description of the quotient map in terms of a reflection group. As an application, one obtains a general description of the branching rules for the restriction of representations of Gl(N) to O(N) andSp(N) as well as detailed information about the structure of the qBrauer algebras in the nonsemisimple case for certain specializations. It is well known that one can study the combinatorics of the finite dimensional representations of the general linear groups Gl(N) in a uniform way essentially independently of the dimension N, using the representation theory of the symmetric groups. A similar approach is possible for orthogonal and symplectic groups. In particular, it is possible to obtain their Grothendieck rings as quotients from a large ring, denoted here by Gr(O(∞)) (see e.g. [KT]). More recently, quotients
QUANTUM GROUPS AND RIBBON GCATEGORIES
, 2001
"... Abstract. For a group G, the notion of a ribbon Gcategory was introduced in [Tu4] with a view towards constructing 3dimensional homotopy quantum field theories (HQFT’s) with target K(G, 1). We discuss here how to derive ribbon Gcategories from a simple complex Lie algebra g where G is the center ..."
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Abstract. For a group G, the notion of a ribbon Gcategory was introduced in [Tu4] with a view towards constructing 3dimensional homotopy quantum field theories (HQFT’s) with target K(G, 1). We discuss here how to derive ribbon Gcategories from a simple complex Lie algebra g where G is the center of g. Our construction is based on a study of representations of the quantum group Uq(g) at a root of unity ε. Under certain assumptions on ε, the resulting Gcategories give rise to numerical invariants of pairs (a closed oriented 3manifold M, an element of H 1 (M; G)) and to 3dimensional HQFT’s. In order to construct 3dimensional homotopy quantum field theories (HQFT’s), the second author introduced for a group G the notions of a ribbon Gcategory and a modular (ribbon) Gcategory. The aim of this paper is to analyze the categories of representations of quantum groups (at roots of unity) from this prospective. The role of G will be played by the center of
Higher Rank TQFT Representations of SL(2, Z) are Reducible Examples of Decompositions and Embeddings.
, 706
"... Abstract: In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum PSU(N) for N> 2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges PSU(N) representation labels ..."
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Abstract: In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum PSU(N) for N> 2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges PSU(N) representation labels by their conjugates. The respective summands of a given parity are typically still reducible into many further components. Specifically, we give an explicit basis for an irreducible direct summand in the SL(2, Z) representation obtained from quantum PSU(3) when the order of the root of unity is a prime r ≡ 2 mod 3. We show that this summand is isomorphic to the respective PSU(2) representation. 1.
Tensor categories: A selective guided tour ∗
, 2008
"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."
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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on klinear categories with finite dimensional homspaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way. 1 Tensor categories 1.1 Strict tensor categories