2017/06/19 by Nampaisarn, Thanasin
#17B10 #17B22 #17B65 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1706.05950
The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category O for the Lie algebra \mathfrakg, where \mathfrakg is one of the finitary, infinite-dimensional Lie algebras \mathfrakgl_∞(\mathbbK), \mathfraksl_∞(\mathbbK), \mathfrakso_∞(\mathbbK), and \mathfraksp_∞(\mathbbK). Here, \mathbbK is an algebraically closed field of characteristic 0. We call these categories "extended categories O" and use the notation O. While the categories O are defined for all splitting Borel subalgebras of \mathfrakg, this research focuses on the categories O for very special Borel subalgebras of \mathfrakg which we call Dynkin Borel subalgebras. Some results concerning block decomposition and Kazhdan-Lusztig multiplicities carry over from usual categories O to our categories 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 O and are able to establish an analogue of BGG reciprocity in the categories O.