Polynomial representations of the Witt Lie algebra

TL;DR

This paper studies polynomial representations of the Witt Lie algebra, proving they are noetherian with rational Hilbert series, and establishes new categorical equivalences and dualities involving these representations.

Contribution

It introduces a novel operadic Schur--Weyl duality and characterizes polynomial representations of W_n through categorical equivalences.

Findings

Polynomial representations of W_n are noetherian.

Polynomial representations have rational Hilbert series.

Equivalences with representations of Fin^op and the endomorphism monoid of A^n.

Abstract

The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor powers of V_n. Our main theorems assert that finitely generated polynomial representations of W_n 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_n 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.