Ordinary and modular representation theory of truncated current Lie algebras

Chaffe, Matthew ORCID: 0000-0001-7650-4633 (2024). Ordinary and modular representation theory of truncated current Lie algebras. University of Birmingham. Ph.D.

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Abstract

In this thesis, we study the representation theory of truncated current Lie algebras associated to Lie algebras of reductive groups. After giving the necessary preliminaries, we begin by considering the representation theory of these Lie algebras in characteristic 0 by defining a generalisation of the Bernstein–Gelfand–Gelfand category \(O\) for reductive Lie algebras and using this to study the problem of computing composition multiplicities
of Verma modules. In particular, we give an inductive procedure to compute these multiplicities in terms of the composition multiplicities of Verma modules for reductive Lie algebras, which are famously given by the Kazhdan–Lusztig polynomials. These results have been published in [7] and [8].

We then move on to consider the representation theory of truncated current Lie algebras in prime characteristic. Here, after proving some elementary structural results such as the classification of semisimple and nilpotent elements, we tackle three main problems. The first is on upper and lower bounds for the dimensions of simple modules; we give an upper bound on the dimensions of simple modules for all \(p\)-characters and a lower bound for certain \(p\)-characters. Then we classify simple modules for certain p-characters. Finally,we finish by computing the composition multiplicities for projective modules for the restricted enveloping algebras of truncated current Lie algebras, and show they can be given in terms of the composition multiplicities of baby Verma modules for the corresponding reductive Lie algebra. These results are the subject of the preprint [9].

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):