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 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 ${⨁}_m{\dot{U}}_q(\mathfrak{gl}(m))$ 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.

Some proofs altered, corrections made as suggested by referees. Published in Algebraic & Geometric topology

Cited by in corpus (3)

• Super $q$-Howe duality and web categories
• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• A topological interpretation of Viro's $\mathfrak{gl}(1|1)$-Alexander polynomial of a graph