Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

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 $\mathfrak{g}=\mathfrak{h}\oplus \cdots \oplus \mathfrak{h}$ (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 $\mathcal{U}(\mathfrak{g})$.

arXiv comment30 pages