A generators and relations description of a representation category of $U_q(\mathfrak{gl}(1|1))$

TL;DR

This paper provides a diagrammatic description of the representation category of $U_q( ext{gl}(1|1))$ using quantum skew Howe duality, revealing new relations and connections to knot invariants like the Alexander polynomial.

Contribution

It introduces a complete diagrammatic framework for the category of $U_q( ext{gl}(1|1))$ representations, including new relations and a link to braid symmetries and knot invariants.

Findings

Diagrammatic description with trivalent graphs and MOY relations

Additional relations specific to the category

Connection to Alexander polynomial construction

Abstract

We use the technique of quantum skew Howe duality to investigate the monoidal category of exterior powers of the standard representation of $U_q(\mathfrak{gl}(1|1))$. This produces a complete diagrammatic description of the category in terms of trivalent graphs, with the usual MOY relations plus one additional family of relations. The technique also gives a useful connection between a system of symmetries on $\bigoplus_m \dot{U}_q(\mathfrak{gl}(m))$ and the braiding on the category of $U_q(\mathfrak{gl}(1|1))$-representations which can be used to construct the Alexander polynomial and coloured variants.