Modified Turaev-Viro invariants from quantum 𝔰𝔩(2|1)

Author: 

Ana-Maria Anghel, C
Geer, N

Publication Date: 

23 March 2020

Journal: 

Journal of Knot Theory and Its Ramifications

Last Updated: 

2020-10-05T13:24:27.687+01:00

Issue: 

04

Volume: 

29

DOI: 

10.1142/s0218216520500182

page: 

2050018-2050018

abstract: 

<jats:p> The category of finite dimensional modules over the quantum superalgebra [Formula: see text] is not semi-simple and the quantum dimension of a generic [Formula: see text]-module vanishes. This vanishing happens for any value of [Formula: see text] (even when [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize [Formula: see text] 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 [Formula: see text]-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures. </jats:p>

Symplectic id: 

1135950

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article