Super duality and homology of unitarizable modules of Lie algebras

TL;DR

This paper derives u-homology formulas for unitarizable modules over classical infinite-rank Lie algebras, recovering Enright's formulas for specific Hermitian symmetric pairs, advancing understanding of their structure and representations.

Contribution

It introduces new u-homology formulas for unitarizable modules at negative levels and extends Enright's formulas to additional classical symmetric pairs.

Findings

Derived u-homology formulas for classical Lie algebras of infinite rank.

Recovered Enright's formulas for specific Hermitian symmetric pairs.

Enhanced understanding of unitarizable modules and their homological properties.

Abstract

The u-homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types gl(n), sp(2n) and so(2n) are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types (SU(p; q); SU(p) X SU(q)), (Sp(2n);U(n)) and (SO*(2n);U(n)).