Gel'fand–Zetlin basis for a class of representations of the Lie superalgebra $gl\left(\infty |\infty \right)$

A new, so called odd Gel'fand–Zetlin (GZ) basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra $gl\left(n|n\right)$. The related GZ patterns are based upon the decomposition according to a particular chain of subalgebras of $gl\left(n|n\right)$. This chain contains only genuine Lie superalgebras of type $gl\left(k|l\right)$ with k and l nonzero (apart from the final element of the chain which is $gl\left(1|0\right)\equiv gl\left(1\right)$). Explicit expressions for a set of generators of the algebra on this GZ 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 $gl\left(\infty |\infty \right)$. (paper)