##### Department of Mathematics,

University of California San Diego

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### Representation Theory Seminar

## Andy Linshaw

#### UCSD

## Invariant chiral differential operators

##### Abstract:

Given a finite-dimensional Lie algebra $g$ and a $g$-module $V$, the ring $D(V)^g$ of invariant differential operators is a much-studied object in classical invariant theory. It has a natural vertex algebra analogue. First, $D(V)$ has a $VA$ analogue $S(V)$ known as a $\beta\gamma$-system or algebra of chiral differential operators. The action of $g$ on $V$ induces an action of the corresponding affine algebra on $S(V)$. The invariant space $S(V)^{g[t]}$ is a commutant subalgebra of $S(V)$, and plays the role of $D(V)^g$. In this talk, I'll describe $S(V)^{g[t]}$ in some basic but nontrivial cases: when $g$ is abelian and the action is diagonalizable, and when $g$ is one of the classical Lie algebras $sl(n), gl(n)$, or so$(n)$, and $V = C^n$. The answer is often a surprise: for example, when $g = C = V, S(V)^{g[t]}$ is the Zamolodchikov $W_3$ algebra with central charge $c=-2$.

### January 29, 2008

### 1:30 PM

### AP&M 7218

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