Some remarks on periodic gradings

TL;DR

This paper investigates the structure of Poisson centers associated with finite order automorphisms of Lie algebras, demonstrating that the Poisson center is polynomial when the algebra is reductive.

Contribution

It introduces a specific Poisson bracket within a family of compatible brackets and proves the polynomiality of the Poisson center for reductive Lie algebras.

Findings

Poisson center ${\\mathcal Z}_\\infty$ is polynomial for reductive Lie algebras

Construction of a large Poisson-commutative subalgebra $Z(\mathfrak q,\vartheta)$

Analysis of a particular Poisson bracket $\\{\,\,\}_{\infty}$ within a pencil of brackets

Abstract

Let $\mathfrak q$ be a finite-dimensional Lie algebra, $\vartheta\in Aut(\mathfrak q)$ a finite order automorphism, and $\mathfrak q_0$ the subalgebra of fixed points of $\vartheta$. Using $\vartheta$ one can construct a pencil $\mathcal P$ of compatible Poisson brackets on $S(\mathfrak q)$, and thereby a `large' Poisson-commutative subalgebra $Z(\mathfrak q,\vartheta)$ consisting of $\mathfrak q_0$-invariants in $S(\mathfrak q)$. We study one particular bracket $\{\,\,,\,\}_{\infty}\in\mathcal P$ and the related Poisson centre ${\mathcal Z}_\infty$. It is shown that ${\mathcal Z}_\infty$ is a polynomial ring, if $\mathfrak q$ is reductive.