Speaker: Andrew Linshaw
Title: Vertex Algebras and Invariant Theory
Abstract: In the theory of vertex algebras, there is a construction known as the commutant which associates to any vertex algebra V and any subalgebra A of V a new subalgebra Com(A,V). This construction was introduced by Frenkel and Zhu in 1992 and generalizes a previous construction due to the physicists Goddard, Kent and Olive known as the coset construction. In general, the problem of describing a commutant subalgebra by giving a list of generators is highly non-trivial. It is not even clear when Com(A,V) is finitely generated as a vertex algebra, even if V and A are finitely generated. In this talk, I will define vertex algebras from scratch and then explain how the commutant construction may be interpreted as a vertex algebra notion of invariant theory. Moreover, I will show how a certain problem in classical invariant theory, namely, the description of the ring of invariant polynomial differential operators on a g-module W, where g is a semisimple Lie algebra, can be "souped up" to a vertex algebra commutant problem. I will discuss some generic features that this commutant algebra possesses for any g and W, and give a complete description in the case where g = sl_2 and W is the adjoint module.