Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of $\mathcal{S}(\mathfrak{g})$

Abstract

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\vartheta \in \mathsf{Aut}(\mathfrak{g})$ a finite order automorphism, and ${\mathfrak{g}}_0$ the subalgebra of fixed points of $\vartheta$. Recently, we noticed that using $\vartheta$ one can construct a pencil of compatible Poisson brackets on $\mathcal{S}(\mathfrak{g})$, and thereby a `large' Poisson-commutative subalgebra $\mathcal{Z}(\mathfrak{g},\vartheta )$ of $\mathcal{S}(\mathfrak{g}{)}^{{\mathfrak{g}}_0}$. In this article, we study invariant-theoretic properties of $(\mathfrak{g},\vartheta )$ that ensure good properties of $\mathcal{Z}(\mathfrak{g},\vartheta )$. Associated with $\vartheta$ one has a natural Lie algebra contraction ${\mathfrak{g}}_{(0)}$ of $\mathfrak{g}$ and the notion of a good generating system (=g.g.s.) in $\mathcal{S}(\mathfrak{g}{)}^{\mathfrak{g}}$. We prove that in many cases the equality $\mathsf{ind}{\mathfrak{g}}_{(0)}=\mathsf{ind}\mathfrak{g}$ holds and $\mathcal{S}(\mathfrak{g}{)}^{\mathfrak{g}}$ has a g.g.s. According to V.G. Kac's classification of finite order automorphisms (1969), $\vartheta$ can be represented by a Kac diagram, $\mathcal{K}(\vartheta )$, and our results often use this presentation. The most surprising observation is that ${\mathfrak{g}}_{(0)}$ depends only on the set of nodes in $\mathcal{K}(\vartheta )$ with nonzero labels, and that if $\vartheta$ is inner and a certain label is nonzero, then ${\mathfrak{g}}_{(0)}$ is isomorphic to a parabolic contraction of $\mathfrak{g}$.

32 pp., to appear in Math. Zeitschrift