On an infinite limit of BGG categories O

TL;DR

This paper investigates an infinite-dimensional version of category O for certain Lie algebras, establishing extension properties, self-duality, and connections to Koszul duality, with implications for understanding infinite root-reductive Lie algebras.

Contribution

It introduces and analyzes a version of category O for infinite root-reductive Lie algebras, revealing extension structures, self-duality, and links to finite-dimensional cases.

Findings

Proved extension fullness and computed higher extensions of simple modules by Verma modules.

Showed that the category O is Ringel self-dual.

Established an equivalence between subquotients of infinite and finite-dimensional category O.

Abstract

We study a version of the BGG category O for Dynkin Borel subalgebras of root-reductive Lie algebras g, such as gl(\infty). We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In addition, we show that our category O is Ringel self-dual and initiate the study of Koszul duality. An important tool in obtaining these results is an equivalence we establish between appropriate Serre subquotients of category O for g and category O for finite dimensional reductive subalgebras of g.