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Topological invariants from nonrestricted quantum groups
, 1009
"... gives rise to the generalized Kashaev and TuraevVirotype 3manifold invariants defined in [12] and [17], respectively. In this case we show that these invariants are equal and extend to what we call a relative Homotopy Quantum Field Theory which is a branch of the Topological Quantum Field Theory ..."
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gives rise to the generalized Kashaev and TuraevVirotype 3manifold invariants defined in [12] and [17], respectively. In this case we show that these invariants are equal and extend to what we call a relative Homotopy Quantum Field Theory which is a branch of the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our main examples of relative spherical categories are the categories of finite dimensional weight modules over nonrestricted quantum groups considered
PatureauMirand  Quantum invariants of 3–manifolds via link surgery presentations and nonsemisimple categories
 Journal of Topology
"... Abstract. In this paper we construct invariants of 3manifolds “a ̀ la ReshetikhinTuraev ” in the setting of nonsemisimple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3–manifold invariants indexed by the integers. We prove this family of invari ..."
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Abstract. In this paper we construct invariants of 3manifolds “a ̀ la ReshetikhinTuraev ” in the setting of nonsemisimple ribbon tensor categories. We give concrete examples of such categories which lead to a family of 3–manifold invariants indexed by the integers. We prove this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard ReshetikhinTuraevWitten invariants do not, and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links.
PatureauMirand  Ambidextrous objects and trace functions for nonsemisimple categories
 Proc. Amer. Math. Soc
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PatureauMirand  Non semisimple TQFTs, Reidemeister torsion and Kashaev’s invariants,arXiv:1404.7289
"... Abstract. We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the nonsemisimple invariants defined in [12] including the Kashaev invariant of links. Here th ..."
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Abstract. We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the nonsemisimple invariants defined in [12] including the Kashaev invariant of links. Here the modular category framework does not apply and we use the “universal construction”. Our TQFT provides a monoidal
Modified traces on Deligne’s category Rep(St
 Journal of Algebraic Combinatorics
"... Abstract. Deligne has defined a category which interpolates among the representations of the various symmetric groups. In this paper we show Deligne’s category admits a unique nontrivial family of modified trace functions. Such modified trace functions have already proven to be interesting in both l ..."
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Abstract. Deligne has defined a category which interpolates among the representations of the various symmetric groups. In this paper we show Deligne’s category admits a unique nontrivial family of modified trace functions. Such modified trace functions have already proven to be interesting in both lowdimensional topology and representation theory. We also introduce a graded variant of Deligne’s category, lift the modified trace functions to the graded setting, and use them to recover the wellknown invariant of framed knots known as the writhe. 1.
IDEALS IN DELIGNE’S TENSOR CATEGORY Rep(GLδ)
"... Abstract. We give a classification of ideals in Rep(GLδ) for arbitrary δ. ..."
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Abstract. We give a classification of ideals in Rep(GLδ) for arbitrary δ.
SOME REMARKS ON THE UNROLLED QUANTUM GROUP OF sl(2)
"... Abstract. In this paper we consider the representation theory of a nonstandard quantization of sl(2). This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In th ..."
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Abstract. In this paper we consider the representation theory of a nonstandard quantization of sl(2). This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In the process of proving these results the paper acts as a survey of the known representation theory associated to this nonstandard quantization of sl(2). The results of this paper are used extensively in [4] to study Topological Quantum Field Theory (TQFT) and have connections with Conformal Field Theory (CFT). 1.