This article is cited in 4 papers

On an infinite limit of BGG categories $\mathcal O$

Kevin Coulembier^a, Ivan Penkov<sup>b

a</sup> School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
^b Jacobs University Bremen, 28759 Bremen, Germany

Abstract: 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 $O$ 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 $\mathfrak{g}$ and category $\mathcal{O}$ for finite dimensional reductive subalgebras of $\mathfrak{g}$.

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

MSC: 17B65, 16S37, 17B55

Language: English

DOI: 10.17323/1609-4514-2019-19-4-655-693

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