# A Hilbert theorem for vertex algebras

Andrew R. Linshaw

Arxiv ID: 0903.3814•Last updated: 8/10/2020

Given a simple vertex algebra A and a reductive group G of automorphisms of
A, the invariant subalgebra A^G is strongly finitely generated in most examples
where its structure is known. This phenomenon is subtle, and is generally not
true of the classical limit of A^G, which often requires infinitely many
generators and infinitely many relations to describe. Using tools from
classical invariant theory, together with recent results on the structure of
the W_{1+\infty} algebra, we establish the strong finite generation of a large
family of invariant subalgebras of \beta\gamma-systems, bc-systems, and
bc\beta\gamma-systems.

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