• Corpus ID: 119120661

# Poisson-commutative subalgebras of $S(\mathfrak g)$ associated with involutions.

@article{Panyushev2018PoissoncommutativeSO,
title={Poisson-commutative subalgebras of \$S(\mathfrak g)\$ associated with involutions.},
author={Dmitri I. Panyushev and Oksana Yakimova},
journal={arXiv: Representation Theory},
year={2018}
}
• Published 2 September 2018
• Mathematics
• arXiv: Representation Theory
The symmetric algebra $S(\mathfrak g)$ of a reductive Lie algebra $\mathfrak g$ is equipped with the standard Poisson structure, i.e., the Lie-Poisson bracket. Poisson-commutative subalgebras of $S(\mathfrak g)$ attract a great deal of attention, because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset S(\mathfrak g)$ is bounded by the "magic number" $\boldsymbol… 5 Citations ## Figures from this paper Reductive subalgebras of semisimple Lie algebras and Poisson commutativity • Mathematics • 2020 Let$\mathfrak g$be a semisimple Lie algebra,$\mathfrak h\subset\mathfrak g$a reductive subalgebra such that$\mathfrak h^\perp$is a complementary$\mathfrak h$-submodule of$\mathfrak g$. In Compatible Poisson brackets associated with 2-splittings and Poisson commutative subalgebras of$\mathcal S(\mathfrak g)$• Mathematics • 2020 Let${\mathcal S}(\mathfrak g)$be the symmetric algebra of a reductive Lie algebra$\mathfrak g$equipped with the standard Poisson structure. If${\mathcal C}\subset\mathcal S(\mathfrak g)$is a Commutative subalgebras of$U(\mathfrak q)$of maximal transcendence degree We prove that the enveloping algebra$U(\mathfrak q)$of a finite-dimensional Lie algebra$\mathfrak q$contains a commutative subalgebra of the maximal possible transcendence degree$(\dim\mathfrak
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