Periodic automorphisms of takiff algebras, contractions, and θ-groups

Abstract

Let $\mathfrak{q}$ be an algebraic Lie algebra and $\mathfrak{q}{\left\langle m \right\rangle }$ a (generalised) Takiff algebra. Any finite-order automorphism θ of $\mathfrak{q}$ induces an automorphism of $\mathfrak{q}{\left\langle m \right\rangle }$ of the same order, denoted ${\hat \theta}$. We study invariant-theoretic properties of representations of the fixed point subalgebra of ${\hat \theta}$ on other eigenspaces of ${\hat \theta}$ in $\mathfrak{q}{\left\langle m \right\rangle }$. We use the observation that, for special values of m, the fixed point subalgebra, $\mathfrak{q}{\left\langle m \right\rangle }^{\hat \theta}$, turns out to be a contraction of a certain Lie algebra associated with $\mathfrak{q}$ and θ.