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Polynomial representations of the Witt Lie algebra

Steven V Sam, Andrew Snowden, Philip Tosteson

TL;DR

This paper establishes finiteness and qualitative growth properties for polynomial representations of the Witt Lie algebra $\mathfrak{W}_n$ by linking them to combinatorial categories. It proves that the category of polynomial $\mathfrak{W}_n$-representations is locally noetherian and that their Hilbert series are rational with denominators built from factors $1-t^m$, with dimensions that are eventually quasi-polynomial. A central insight is that polynomial $\mathfrak{W}$-modules correspond to $\mathbf{Fin}^{\mathrm{op}}$-modules, enabling transfer of noetherianity and Hilbert-series results via specialization functors and Serre quotients. The work also develops a broad operadic framework showing an equivalence between polynomial representations of derivation Lie algebras and modules over wiring categories, generalizing Schur–Weyl duality and connecting to $\mathcal{W}$- and $P$-algebra structures. By weaving together operads, wiring categories, and $\mathbf{FS}^{\mathrm{op}}$-modules, the results illuminate finiteness phenomena in infinite-dimensional Lie algebra representations and give tools for representation stability and related combinatorial algebra problems.

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.

Polynomial representations of the Witt Lie algebra

TL;DR

This paper establishes finiteness and qualitative growth properties for polynomial representations of the Witt Lie algebra by linking them to combinatorial categories. It proves that the category of polynomial -representations is locally noetherian and that their Hilbert series are rational with denominators built from factors , with dimensions that are eventually quasi-polynomial. A central insight is that polynomial -modules correspond to -modules, enabling transfer of noetherianity and Hilbert-series results via specialization functors and Serre quotients. The work also develops a broad operadic framework showing an equivalence between polynomial representations of derivation Lie algebras and modules over wiring categories, generalizing Schur–Weyl duality and connecting to - and -algebra structures. By weaving together operads, wiring categories, and -modules, the results illuminate finiteness phenomena in infinite-dimensional Lie algebra representations and give tools for representation stability and related combinatorial algebra problems.

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.
Paper Structure (40 sections, 34 theorems, 64 equations)

This paper contains 40 sections, 34 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Main results
  4. Overview of proof
  5. An operadic generalization
  6. Motivation
  7. Open problems
  8. Relation to other work
  9. Outline
  10. Background
  11. Polynomial representations
  12. Operads
  13. Derivations
  14. Wiring categories (PROPs)
  15. Some examples
  16. ...and 25 more sections

Key Result

Theorem 1.3

If $M$ is a finitely generated polynomial representation of $\mathfrak{W}_n$ then any subrepresentation of $M$ is also finitely generated.

Theorems & Definitions (78)

  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 2.1
  • ...and 68 more