Generalized Casimir Operators for Lie Superalgebras

TL;DR

This paper introduces generalized Casimir operators for loop contragredient Lie superalgebras, proving their commutation properties and applications in tensor product module decompositions, along with generalized Gelfand invariants.

Contribution

It defines new generalized Casimir operators and Gelfand invariants for Lie superalgebras, expanding tools for representation theory analysis.

Findings

Generalized Casimir operators commute with the Lie superalgebra.

These operators can generate new highest weight vectors in tensor products.

Introduction of generalized Gelfand invariants for loop general linear Lie superalgebras.

Abstract

In this paper, we define generalized Casimir operators for a loop contragredient Lie superalgebra and prove that they commute with the underlying Lie superalgebra. These operators have applications in the decomposition of tensor product modules. We further introduce the notion of generalized Gelfand invariants for the loop general linear Lie superalgebra and show that they also commute with the underlying Lie superalgebra. These operators when applied to a highest weight vector in a tensor product module again induces a new highest weight vector.