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Abstract. We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl 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.

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional 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.

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 BMW-algebras. 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.

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 Witten-Reshetikhin-Turaev invariant τ Pg for closed 3-manifolds. 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.

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 level-rank duality, Kirby-Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves G2 and does not appear to be related to level-rank duality. AMS Classification 18D10; 57M27 17B10 81R05 57R56

Abstract. For a group G, the notion of a ribbon G-category was introduced in [Tu4] with a view towards constructing 3-dimensional homotopy quantum field theories (HQFT’s) with target K(G, 1). We discuss here how to derive ribbon G-categories 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 G-categories give rise to numerical invariants of pairs (a closed oriented 3-manifold M, an element of H 1 (M; G)) and to 3-dimensional HQFT’s. In order to construct 3-dimensional homotopy quantum field theories (HQFT’s), the second author introduced for a group G the notions of a ribbon G-category and a modular (ribbon) G-category. 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

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.