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

Abstract

We use the technique of quantum skew Howe duality to investigate the monoidal category generated by exterior powers of the standard representation of $U_q\left(\mathfrak{g}\mathfrak{l}\left(1|1\right)\right)$. 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 ${\oplus }_m{\dot{U}}_q\left(\mathfrak{g}\mathfrak{l}\left(m\right)\right)$ and the braiding on the category of $U_q\left(\mathfrak{g}\mathfrak{l}\left(1|1\right)\right)$–representations which can be used to construct the Alexander polynomial and coloured variants.