Quasifinite representations of the Lie superalgebra of quantum pseudo differential operators

TL;DR

This paper extends classification results for quasifinite highest weight modules from $ ext{Z}$-graded Lie algebras to ${1/2} ext{Z}$-graded Lie superalgebras, specifically applied to quantum pseudo-differential operators.

Contribution

It generalizes existing theories to superalgebras and classifies irreducible quasifinite highest weight modules for the Lie superalgebra of quantum pseudo-differential operators.

Findings

Extended classification to ${1/2}\Z$-graded Lie superalgebras.

Classified irreducible quasifinite highest weight modules.

Applied results specifically to quantum pseudo-differential operators.

Abstract

In this paper we extend general results obtained by V. Kac and J. Liberati, in "Unitary quasifinite representations of $W_\infty$", (Letters Math. Phys., 53 (2000), 11-27), for quasifinite highest weight representations of $\Z$-graded Lie algebras to ${1/2}\Z$-graded Lie superalgebras, and we apply these to classify the irreducible quasifinite highest weight modules of the Lie superalgebra of quantum pseudo-differential operators.