Compatible Poisson brackets associated with 2-splittings and Poisson commutative subalgebras of $\mathcal S(\mathfrak g)$

DOI10.1112/JLMS.12418arXiv2009.05271MaRDI QIDQ6348879

Oksana S. Yakimova, Dmitri I. Panyushev

Publication date: 11 September 2020

Abstract: Let

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

be the symmetric algebra of a reductive Lie algebra

m a t h f r a k g

equipped with the standard Poisson structure. If

m a t h c a l C s u b s e t m a t h c a l S (m a t h f r a k g)

is a Poisson-commutative subalgebra, then

, where

. We present a method for constructing the Poisson-commutative subalgebra

m a t h c a l Z_{l a n g l e m a t h f r a k h, m a t h f r a k r a n g l e}

of transcendence degree

via a vector space decomposition

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

into a sum of two spherical subalgebras. There are some natural examples, where the algebra

m a t h c a l Z_{l a n g l e m a t h f r a k h, m a t h f r a k r a n g l e}

appears to be polynomial. The most interesting case is related to the pair

(m a t h f r a k b, m a t h f r a k u_{-})

, where

m a t h f r a k b

is a Borel subalgebra of

m a t h f r a k g

. Here we prove that

{m a t h c a l Z}_{l a n g l e m a t h b b b, m a t h b b u_{-} a n g l e}

is maximal Poisson-commutative and is complete on every regular coadjoint orbit in

m a t h f r a k g^{*}

. Other series of examples are related to decompositions associated with involutions of

m a t h f r a k g

.

Mathematics Subject Classification ID

Group actions on varieties or schemes (quotients) (14L30) Semisimple Lie groups and their representations (22E46) Poisson algebras (17B63) Simple, semisimple, reductive (super)algebras (17B20) Coadjoint orbits; nilpotent varieties (17B08)

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