Mr Lewis Topley (University of Birmingham) gives a maths seminar at the School of Mathematics, UEA. The talk is on “The dimensions of modules for Lie algebras in positive characteristics”. The abstract is below.

One of the main goals in representation theory is to understand simple modules for a chosen class of algebraic objects. Lie algebras are one of the most classical algebraic structures: they arise naturally as the infinitesimal analogues of (continuous) groups. Over the complex numbers representations of Lie algebras have been studied extensively and the situation is quite well-understood, although many mysteries still remain. Over fields of positive characteristics however, the situation is more complicated. I will begin this talk by giving a gentle introduction to this field, comparing the ordinary and the modular.

In the second half of the talk I will describe a joint work with Ben Martin and David Stewart in which we apply the Leftschetz principle, along with classical techniques from Lie theory, to prove (a generic version of) a conjecture of Kac and Weisfeiler from 1971, describing the maximal dimension of simple modules over a Lie algebra in positive characteristic.