PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

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