Linear Lie algebras with finite dimensional centralizers

TL;DR

This paper establishes criteria to determine when the polynomial centralizer of a linear Lie algebra is finite or infinite dimensional, linking it to group invariants, and explores applications to normal forms and fundamental solutions.

Contribution

It provides new criteria based on invariants for the dimensionality of polynomial centralizers in linear Lie algebras, with applications to normal forms and differential equations.

Findings

Criteria for finite vs. infinite dimensional polynomial centralizers

Connections between invariants and centralizer properties

Applications to normal forms and fundamental solutions

Abstract

We give criteria for finite dimensionality or infinite dimensionality of the polynomial centralizer of the Lie algebra of a linear Lie group, in terms of invariants and relative invariants of the group. In the finite dimensional scenario some applications to normal forms and to certain equations with fundamental solutions are presented.