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

The category of finite dimensional modules over the quantum superalgebra Uq𝔰𝔩(2|1) is not semi-simple and the quantum dimension of a generic Uq𝔰𝔩(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 Uq𝔰𝔩(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.