Howe duality and Kostant Homology Formula for infinite-dimensional Lie superalgebras

TL;DR

This paper uses Howe duality to explicitly compute Kostant-type homology groups for representations of infinite-dimensional Lie superalgebras and their subalgebras, providing new formulas and resolutions.

Contribution

It introduces explicit Kostant homology formulas for $\,rak{gl}_{infty|infty}$ and its subalgebras at various levels, and constructs resolutions with generalized Verma modules.

Findings

Computed Kostant homology groups explicitly for infinite-dimensional Lie superalgebras.

Derived Kostant-type homology formulas at positive and negative integral levels.

Constructed resolutions using generalized Verma modules.

Abstract

Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra $\hat{\frak{gl}}_{\infty|\infty}$ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra $ widehat{\frak{gl}}_\infty$ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations.