JournalsjemsVol. 13, No. 6pp. 1737–1768

Invariant theory and the W1+\mathcal{W}_{1+\infty} algebra with negative integral central charge

  • Andrew R. Linshaw

    Technische Hochschule Darmstadt, Germany
Invariant theory and the $\mathcal{W}_{1+\infty}$ algebra with negative integral central charge cover
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Abstract

The vertex algebra W1+,c\mathcal{W}_{1+\infty,c} with central charge cc may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n1n\geq 1, it was conjectured in the physics literature that W1+,n\mathcal{W}_{1+\infty,-n} should have a minimal strong generating set consisting of n2+2nn^2+2n elements. Using a free field realization of W1+,n\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 GLnGL_n, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of W1+,n\mathcal{W}_{1+\infty,-n} are parametrized by a closed subvariety of Cn2+2n\mathbb{C}^{n^2+2n}.

Cite this article

Andrew R. Linshaw, Invariant theory and the W1+\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