Polynomial Representations of the Witt Lie Algebra

The Witt algebra Wₙ is the Lie algebra of all derivations of the n-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.