### Seminar in Algebra, Lie Theory, and Geometry

# Thanasin Nampaisarn

### (Jacobs University Bremen)

## "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.