Projective modules over classical Lie algebras of infinite rank in the parabolic category

TL;DR

This paper investigates the structure of projective modules in the parabolic category O over infinite rank classical Lie algebras and superalgebras, establishing their Koszulity and the existence of projective covers.

Contribution

It proves the existence of projective covers for all irreducible modules and shows that the category O is Koszul for both Lie algebras and superalgebras of types A, B, C, D.

Findings

Existence of projective covers for irreducible modules in category O.

Category O is proven to be a Koszul category.

Extension of results to superalgebras via super duality.

Abstract

We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.