Invariant theory and the W_{1+\infty} algebra with negative integral central charge

Details Invariant theory and the W_{1+\infty} algebra with negative integral central charge Invariant theory and the W_{1+\infty} algebra with negative integral central charge

2008-11-25

arxiv.org/abs/0811.4067v6

J. Eur. Math. Soc. vol. 13, no. 6 (2011) 1737-1768

Mathematics

Representation Theory

Minor corrections, final version

Scientific paper

The vertex algebra 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 W_{1+\infty,-n} should have a minimal strong generating set consisting of n^2+2n elements. Using a free field realization of 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 W_{1+\infty,-n} are parametrized by a closed subvariety of C^{n^2+2n}.

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