#### Volume 16, issue 1 (2016)

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

### Jonathan Grant

Algebraic & Geometric Topology 16 (2016) 509–539
##### 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 ${\oplus }_{m}{\stackrel{̇}{U}}_{q}\left(\mathfrak{g}\mathfrak{l}\left(m\right)\right)$ 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.

##### Keywords
skew Howe duality, diagram calculus, knot polynomial, quantum group
Primary: 17B37
Secondary: 57M25