On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings

TL;DR

This paper extends the understanding of finite-dimensional Lie algebras generated by homogeneous derivations of polynomial rings, including those not locally nilpotent, providing new criteria for their finite dimensionality.

Contribution

It introduces a finite dimensionality criterion for Lie algebras generated by homogeneous derivations that are not necessarily locally nilpotent, expanding previous results.

Findings

Finite dimensionality criterion for non-locally nilpotent derivations.

Characterization of Lie algebra structures generated by these derivations.

Extension of known results from locally nilpotent to general homogeneous derivations.

Abstract

For a finite set of homogeneous locally nilpotent derivations of the algebra of polynomials in several variables, a finite dimensionality criterion for the Lie algebra generated by these derivations is known. Also the structure of the corresponding finite-dimensional Lie algebras is described in previous works. In this paper, we obtain a finite dimensionality criterion for a Lie algebra generated by a finite set of homogeneous derivations, each of which is not locally nilpotent.