# Invariant theory and the $\mathcal{W}_{1+\infty}$ algebra with negative integral central charge

### Andrew R. Linshaw

Technische Hochschule Darmstadt, Germany

## Abstract

The vertex algebra $\mathcal{W}_{1+\infty,c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\geq 1$, it was conjectured in the physics literature that $\mathcal{W}_{1+\infty,-n}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of $\mathcal{W}_{1+\infty,-n}$ due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of $GL_n$, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of $\mathcal{W}_{1+\infty,-n}$ are parametrized by a closed subvariety of $\mathbb{C}^{n^2+2n}$.

## Cite this article

Andrew R. Linshaw, Invariant theory and the $\mathcal{W}_{1+\infty}$ algebra with negative integral central charge. J. Eur. Math. Soc. 13 (2011), no. 6, pp. 1737–1768

DOI 10.4171/JEMS/292