An invariant supertrace for the category of representations of Lie superalgebras

TL;DR

This paper introduces a re-normalized supertrace for Lie superalgebra representations that remains non-trivial and invariant, overcoming the limitations of traditional superdimensions and supertraces.

Contribution

It provides a new invariant supertrace for type I Lie superalgebras using a modified superdimension, connecting classical Lie theory with quantum algebra and topology.

Findings

Modified superdimensions are non-zero and lead to a non-trivial supertrace.

The new supertrace induces a non-zero bilinear form on invariant tensor spaces.

Classical results are proved using quantum algebra and topology techniques.

Abstract

In this paper we give a re-normalization 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 non-zero and lead to a kind of supertrace which is non-trivial and invariant. As an application we show that this new supertrace gives rise to a non-zero 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 can not prove them without using quantum algebra and low-dimensional topology.