The half-twist for Uq(g) representations

Snyder, Noah; Tingley, Peter

doi:10.2140/ant.2009.3.809

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The half-twist for $U_q(\mathfrak{g})$ representations

Noah Snyder and Peter Tingley

Vol. 3 (2009), No. 7, 809–834

DOI: 10.2140/ant.2009.3.809

Abstract

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra $ℋ$ along with a distinguished element $t \in ℋ$ such that $(ℋ, R, C)$ is a ribbon Hopf algebra, where $R = (t^{- 1} \otimes t^{- 1}) Δ (t)$ and $C = t^{- 2}$ . The element $t$ is closely related to the topological “half-twist”, which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the category of representations of any half-ribbon Hopf algebra. We show that $U_{q} (g)$ is a (topological) half-ribbon Hopf algebra, but that $t^{- 2}$ is not the standard ribbon element. For $U_{q} ({s l}_{2})$ , we show that there is no half-ribbon element $t$ such that $t^{- 2}$ is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.

Keywords

quantum group, Hopf algebra, ribbon category

Mathematical Subject Classification 2000

Primary: 17B37

Secondary: 57T05, 57M05

Milestones

Received: 6 October 2008

Revised: 11 September 2009

Accepted: 17 October 2009

Published: 29 November 2009

Authors

Noah Snyder
Department of Mathematics Columbia University Rm 626, MC 4403 2990 Broadway New York, NY 10027 United States
http://math.columbia.edu/~nsnyder/
Peter Tingley
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Ave Cambridge, MA 02139 United States
http://www-math.mit.edu/~ptingley/

Vol. 3, No. 7, 2009