Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

TL;DR

This paper explores how finite order automorphisms of semisimple Lie algebras induce compatible Poisson brackets and maximal Poisson-commutative subalgebras, which are quantized via Gaudin subalgebras.

Contribution

It establishes a connection between automorphisms, compatible Poisson structures, and Gaudin subalgebras, providing new polynomial maximal subalgebras for specific Lie algebra decompositions.

Findings

Poisson-commutative subalgebra is polynomial and maximal for cyclic permutations of repeated simple Lie algebras.

Constructs a quantization of these subalgebras using Gaudin models.

Utilizes Vinberg's theory and invariant theory for semisimple Lie algebras.

Abstract

Let $\mathfrak g$ be a finite-dimensional Lie algebra. The symmetric algebra $\mathcal S(\mathfrak g)$ is equipped with the standard Lie-Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on $\mathcal S(\mathfrak g)$ to any finite order automorphism $\theta$ of $\mathfrak g$. We study related Poisson-commutative subalgebras $\mathcal C$ of $\mathcal S(\mathfrak g)$ and associated Lie algebra contractions of $\mathfrak g$. To obtain substantial results, we have to assume that $\mathfrak g$ is semisimple. Then we can use Vinberg's theory of $\theta$-groups and the machinery of Invariant Theory. If $\mathfrak g=\mathfrak h\oplus\dots \oplus \mathfrak h$ (sum of $k$ copies), where $\mathfrak h$ is simple, and $\theta$ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra $\mathcal C$ is polynomial and maximal. Furthermore, we quantise this $\mathcal C$ using a Gaudin subalgebra in the enveloping algebra $\mathcal U(\mathfrak g)$.