An invariant supertrace for the category of representations of Lie superalgebras of type I

Nathan Geer and Bertrand Patureau-Mirand

Vol. 238 (2008), No. 2, 331–348

DOI: 10.2140/pjm.2008.238.331

Abstract

In this paper we give a renormalization of the supertrace on the category of representations of Lie superalgebras of type I, by a kind of modified superdimension. The genuine superdimensions and supertraces are generically zero. However, these modified superdimensions are nonzero and lead to a kind of supertrace which is nontrivial and invariant. As an application we show that this new supertrace gives rise to a nonzero bilinear form on a space of invariant tensors of a Lie superalgebra of type I. The results of this paper are completely classical results in the theory of Lie superalgebras but surprisingly we cannot prove them without using quantum algebra and low-dimensional topology.

Keywords

Lie superalgebra, tensor category, quantum group, knot, trace

Mathematical Subject Classification 2000

Primary: 17B99

Secondary: 17B37

Milestones

Received: 10 December 2007

Revised: 24 June 2008

Accepted: 17 July 2008

Published: 1 December 2008