Periodic automorphisms of Takiff algebras, contractions, and $\theta$-groups

TL;DR

This paper investigates automorphisms of Takiff algebras derived from algebraic Lie algebras, focusing on their invariant theory and the properties of fixed point subalgebras under these automorphisms.

Contribution

It introduces a framework for understanding automorphisms of Takiff algebras and analyzes their invariant-theoretic properties, extending the theory of automorphisms to generalized Takiff structures.

Findings

Automorphisms of Takiff algebras of finite order are induced by automorphisms of the original Lie algebra.

Invariant theory of fixed point subalgebras under these automorphisms is systematically studied.

Results connect automorphisms, Takiff algebras, and invariant theory in a unified framework.

Abstract

Let $q$ be an algebraic Lie algebra and $q<m>$ a (generalised) Takiff algebra. Any finite order automorphism $\theta$ of $q$ induces an automorphisms of $q<m>$ of the same order, denoted $\Theta$. We study invariant-theoretic properties of representations of the fixed point subalgebra of $\Theta$ on other eigenspaces of $\Theta$ in $q<m>$.