Modified Turaev-Viro Invariants from Quantum sl(2|1)

Document Type

Article

Journal/Book Title/Conference

Journal of Knot Theory and Its Ramifications

Volume

29

Issue

4

Publisher

World Scientific Publishing Co Pte Ltd

Publication Date

3-23-2020

Award Number

NSF, Division of Mathematical Sciences (DMS) 1664387

Funder

NSF, Division of Mathematical Sciences (DMS)

Abstract

The category of finite dimensional modules over the quantum superalgebra Uq����(2|1) is not semi-simple and the quantum dimension of a generic Uq����(2|1)-module vanishes. This vanishing happens for any value of q (even when q is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize q to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of Uq����(2|1) by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize q to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative G-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

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