Trace-like invariant for representations of nilpotent liftings of quantum planes

TL;DR

This paper derives a formula for the trace of the antipode on simple modules of nilpotent liftings of quantum planes, linking it to the quantum dimension up to a scalar, advancing understanding of quantum algebra invariants.

Contribution

It provides a new explicit formula for the trace of the antipode on simple modules in this specific quantum algebra setting.

Findings

Trace of the antipode equals quantum dimension times a scalar.

The scalar depends non-trivially on the simple module.

The formula applies to nilpotent liftings of quantum planes.

Abstract

We derive a formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent liftings of quantum planes. We show that the trace is equal to the quantum dimension of the module up to a nonzero scalar depending on the simple module.