Periodic automorphisms of Takiff algebras, contractions, and $\theta$-groups
Let $q$ be an algebraic Lie algebra and $q $ a (generalised) Takiff algebra. Any finite order automorphism $\theta$ of $q$ induces an automorphisms of $q $ 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 $.