Topological invariants from non-restricted quantum groups

TL;DR

This paper introduces relative spherical categories from non-restricted quantum groups, demonstrating their role in defining generalized 3-manifold invariants and extending topological quantum field theories.

Contribution

It establishes a new class of categories called relative spherical categories and connects them to generalized 3-manifold invariants and topological quantum field theories.

Findings

Relative spherical categories yield equivalent Kashaev and Turaev-Viro invariants.

Categories of finite-dimensional weight modules over non-restricted quantum groups are examples.

These categories produce ribbon categories and re-normalized link invariants, including Alexander-type invariants.

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.