Polynomial Representations of the Witt Lie Algebra

The Witt algebra Wn is the Lie algebra of all derivations of then-variable polynomial ring Vn=C[x1,...,xn] (or of algebraic vector fields on An). A representation of Wn is polynomial if it arises as a subquotient of a sum of tensor powers of Vn. Our main theorems assert that finitely generated polynomial representations of Wn are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of Finop, where Fin is the category of finite sets. We also show that polynomial representations of Wn are equivalent to polynomial representations of the endomorphism monoid of An. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.