A generators and relations description of a representation category of Uq(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 Uq(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 ⊕m Uq(gl(m)) and the braiding on the category of Uq(gl(1|1))-representations which can be used to construct the Alexander polynomial and coloured variants.