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 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 mU̇q(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.

Keywords
skew Howe duality, diagram calculus, knot polynomial, quantum group
Mathematical Subject Classification 2010
Primary: 17B37
Secondary: 57M25
References
Publication
Received: 2 December 2014
Revised: 13 May 2015
Accepted: 6 July 2015
Published: 23 February 2016
Authors
Jonathan Grant
Department of Mathematical Sciences
Durham University
Science Laboratories
South Rd.
Durham DH1 3LE
UK
http://www.maths.dur.ac.uk/users/jonathan.grant/