Modified Turaev-Viro Invariants from Quantum sl(2|1)
Journal of Knot Theory and Its Ramifications
World Scientific Publishing Co Pte Ltd
NSF, Division of Mathematical Sciences (DMS) 1664387
NSF, Division of Mathematical Sciences (DMS)
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.
Ana-Maria Anghel, Cristina, and Nathan Geer. “Modified Turaev-Viro Invariants from Quantum 𝔰𝔩(2|1).” Journal of Knot Theory and Its Ramifications, vol. 29, no. 04, Apr. 2020, p. 2050018. DOI.org (Crossref), https://doi.org/10.1142/S0218216520500182.