Polynomial Representations of the Witt Lie Algebra

Abstract The Witt algebra W₍ is the Lie algebra of all derivations of the n-variable polynomial ring V₍=Cx₁, , x₍ (or of algebraic vector fields on A^n). A representation of W₍ is polynomial if it arises as a subquotient of a sum of tensor powers of V₍. Our main theorems assert that finitely generated polynomial representations of W₍ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of Fin^op, where Fin is the category of finite sets. We also show that polynomial representations of W₍ are equivalent to polynomial representations of the endomorphism monoid of A^n. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.