Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

DOI10.1007/S00031-021-09650-3arXiv2102.10065MaRDI QIDQ6361080

Oksana S. Yakimova, Dmitri Panyushev

Publication date: 19 February 2021

Abstract: Let

m a t h f r a k g

be a finite-dimensional Lie algebra. The symmetric algebra

m a t h c a l S (m a t h f r a k 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

m a t h c a l S (m a t h f r a k g)

to any finite order automorphism

h e t a

of

m a t h f r a k g

. We study related Poisson-commutative subalgebras

m a t h c a l C

of

m a t h c a l S (m a t h f r a k g)

and associated Lie algebra contractions of

m a t h f r a k g

. To obtain substantial results, we have to assume that

m a t h f r a k g

is semisimple. Then we can use Vinberg's theory of

h e t a

-groups and the machinery of Invariant Theory. If

m a t h f r a k g = m a t h f r a k h o p l u s d o t s o p l u s m a t h f r a k h

(sum of

k

copies), where

m a t h f r a k h

is simple, and

h e t a

is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra

m a t h c a l C

is polynomial and maximal. Furthermore, we quantise this

m a t h c a l C

using a Gaudin subalgebra in the enveloping algebra

m a t h c a l U (m a t h f r a k g)

.

Mathematics Subject Classification ID

Semisimple Lie groups and their representations (22E46) Poisson algebras (17B63) Simple, semisimple, reductive (super)algebras (17B20) Coadjoint orbits; nilpotent varieties (17B08)

This page was built for publication: Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6361080)