Topological invariants from non-restricted quantum groups

Abstract

We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, 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 non-restricted quantum groups considered by C. De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories are not semi-simple and have an infinite number of non-isomorphic irreducible modules all having vanishing quantum dimensions. We also show that these categories have associated ribbon categories which gives rise to re-normalized link invariants. In the case of sl(2) these link invariants are the Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.

37 pages, 16 figures

References in corpus (2)

• Modified quantum dimensions and re-normalized link invariants
• Quantum hyperbolic geometry

Cited by in corpus (9)

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• Non-Semisimple Extended Topological Quantum Field Theories
• Modified graded Hennings invariants from unrolled quantum groups and modified integral
• Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
• Categories of Weight Modules for Unrolled Restricted Quantum Groups at Roots of Unity