Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

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 $\mathcal U(\mathfrak g)$.