# Moscow Mathematical Journal

Volume 19, Issue 4, October–December 2019  pp. 655–693.

On an Infinite Limit of BGG Categories $\mathcal{O}$

Authors:  Kevin Coulembier (1) and Ivan Penkov (2)
Author institution:(1) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
(2) Jacobs University Bremen, 28759 Bremen, Germany

Summary:

We study a version of the BGG category $\mathcal{O}$ for Dynkin Borel subalgebras of root-reductive Lie algebras $\mathfrak{g}$, such as $\mathfrak{gl}(\infty)$. We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In addition, we show that our category $\mathbf{O}$ 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 $\mathbf{O}$ for $\mathfrak{g}$ and category $\mathcal{O}$ for finite dimensional reductive subalgebras of $\mathfrak{g}$.

2010 Math. Subj. Class. 17B65, 16S37, 17B55.

Keywords: BGG Category $\mathcal{O}$, root-reductive Lie algebra, Dynkin Borel subalgebra, Koszul duality, Ringel duality, Verma module, Serre subquotient category, quasi-hereditary algebra.