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.