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

TL;DR

This paper explores how finite order automorphisms of semisimple Lie algebras influence Poisson-commutative subalgebras, contractions, and invariant properties, revealing structural dependencies on automorphism diagrams.

Contribution

It establishes conditions under which Poisson-commutative subalgebras have good properties and relates automorphism diagrams to algebra contractions and invariants.

Findings

Equality of indices for certain contractions and original algebras.

Existence of good generating systems in invariant subalgebras.

Dependence of contractions on automorphism diagram nodes.

Abstract

Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf Aut}(\mathfrak g)$ 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 $\mathfrak g$.