Date:	Tue, June 6, 2017
Time:	13:00
Place:	IRC Seminar Room

Abstract: The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category \(\mathcal{O}\) for the Lie algebra \(\mathfrak{g}\), where \(\mathfrak{g}\) is one of the finitary, infinite-dimensional Lie algebras \(\mathfrak{gl}_\infty(\mathbb{K})\), \(\mathfrak{sl}_\infty(\mathbb{K})\), \(\mathfrak{so}_\infty(\mathbb{K})\), and \(\mathfrak{sp}_\infty(\mathbb{K})\). Here, \(\mathbb{K}\) is an algebraically closed field of characteristic \(0\). We call these categories extended categories \(\mathcal{O}\) and use the notation \(\bar{\mathcal{O}}\). While the categories \(\bar{\mathcal{O}}\) are defined for all splitting Borel subalgebras of \(\mathfrak{g}\), this research focuses on the categories \(\bar{\mathcal{O}}\) for very special Borel subalgebras of \(\mathfrak{g}\) which we call Dynkin Borel subalgebras. Some results concerning block decomposition and Kazhdan-Lusztig multiplicities carry over from usual categories \(\mathcal{O}\) to our categories \(\bar{\mathcal{O}}\). There are differences which we shall explore in detail, such as the lack of some injective hulls. In this connection, we study truncated categories \(\bar{\mathcal{O}}\) and are able to establish an analogue of BGG reciprocity in the categories \(\bar{\mathcal{O}}\).