Gel'fand-Zetlin basis for a class of representations of the Lie superalgebra gl(\infty|\infty)

TL;DR

This paper introduces an odd Gel'fand-Zetlin basis for irreducible covariant tensor representations of the Lie superalgebra gl(n|n), providing explicit generator expressions and extending to modules of gl(|).

Contribution

It presents a novel basis for gl(n|n) representations and constructs a class of irreducible modules for gl(|), advancing the understanding of Lie superalgebra representations.

Findings

Defined the odd Gel'fand-Zetlin basis for gl(n|n)

Derived explicit generator expressions on this basis

Extended results to modules of gl(|)

Abstract

A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(n|n). The related Gel'fand-Zetlin patterns are based upon the decomposition according to a particular chain of subalgebras of gl(n|n). This chain contains only genuine Lie superalgebras of type gl(k|l) with k and l nonzero (apart from the final element of the chain which is gl(1|0)=gl(1)). Explicit expressions for a set of generators of the algebra on this Gel'fand-Zetlin basis are determined. The results are extended to an explicit construction of a class of irreducible highest weight modules of the general linear Lie superalgebra gl(\infty|\infty).