# Thanasin Nampaisarn

## "Extended BGG Categories $$\mathcal{O}$$ for Locally Semisimple Finitary Lie Algebras"

 Date: Thu, February 16, 2017 Time: 11:30 Place: Seminar Room (120), Research I

Abstract: In this talk, I shall give an overview of my PhD research. If $$\mathfrak{g}$$ is a locally semisimple finitary Lie algebra with a splitting Borel subalgebra $$\mathfrak{b}$$ containing a splitting maximal toral subalgebra $$\mathfrak{h}$$, then the extended BGG category $$\mathcal{O}$$, denoted by $$\bar{\mathcal{O}}^\mathfrak{g}_\mathfrak{b}$$, is the full subcategory of the category of $$\mathfrak{g}$$-modules consisting of locally $$\mathfrak{n}$$-finite $$\mathfrak{h}$$-weight modules with finite-dimensional weight spaces, where $$\mathfrak{n}$$ is the locally nilpotent subalgebra such that $$\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}$$. This infinite-dimensional analogue $$\bar{\mathcal{O}}^\mathfrak{g}_\mathfrak{b}$$ of the classical BGG category $$\mathcal{O}$$ offers both similarities to and crucial differences with the finite-dimensional theory. This talk shall focus on some of these differences.