Vol. 3, No. 7, 2009

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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

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra along with a distinguished element t such that (,R,C) is a ribbon Hopf algebra, where R = (t1 t1)Δ(t) and C = t2. 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 Uq(g) is a (topological) half-ribbon Hopf algebra, but that t2 is not the standard ribbon element. For Uq(sl2), we show that there is no half-ribbon element t such that t2 is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.

quantum group, Hopf algebra, ribbon category
Mathematical Subject Classification 2000
Primary: 17B37
Secondary: 57T05, 57M05
Received: 6 October 2008
Revised: 11 September 2009
Accepted: 17 October 2009
Published: 29 November 2009
Noah Snyder
Department of Mathematics
Columbia University
Rm 626, MC 4403
2990 Broadway
New York, NY 10027
United States
Peter Tingley
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Ave
Cambridge, MA 02139
United States