A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

Grant, Jonathan

doi:10.2140/agt.2016.16.509

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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

DOI: 10.2140/agt.2016.16.509

Abstract

We use the technique of quantum skew Howe duality to investigate the monoidal category generated by exterior powers of the standard representation of $U_{q} (g l (1 | 1))$ . 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 $\oplus_{m} {\dot{U}}_{q} (g l (m))$ and the braiding on the category of $U_{q} (g l (1 | 1))$ –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/

Volume 16, issue 1 (2016)