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Generalized Casimir Operators for Lie Superalgebras

S. Eswara Rao

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.

Generalized Casimir Operators for 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.

Paper Structure

This paper contains 5 sections, 14 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Contragredient Lie Superalgebras and Casimir Operator
  3. Casimir Operators and Loop Algebras
  4. Tensor Product Modules For General Lie Superalgebras
  5. Construction Of Anti-invariants

Key Result

Theorem 2.1

If $V \in \mathcal{O}$, then $(\Omega X - X \Omega).V=0 \ \forall \ X \in \mathfrak{g}$.

Theorems & Definitions (25)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • Proposition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 15 more