Volume 16, issue 1 (2016)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
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
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/