# Brandeis University Everyperson Seminar

## Monday December 6, 2004, 4 PM, Goldsmith 317

### tea at
3:30 PM at the Math Department lounge (Goldsmith 300)

**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.