Poissoncommutative subalgebras of $S(\mathfrak g)$ associated with involutions
Abstract
The symmetric algebra $S(\mathfrak g)$ of a reductive Lie algebra $\mathfrak g$ is equipped with the standard Poisson structure, i.e., the LiePoisson bracket. Poissoncommutative 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 Poissoncommutative subalgebra ${\mathcal C}\subset S(\mathfrak g)$ is bounded by the "magic number" $\boldsymbol{b}(\mathfrak g)$ of $\mathfrak g$. The "argument shift method" of MishchenkoFomenko was basically the only known source of $\mathcal C$ with ${\rm trdeg\,}{\mathcal C}=\boldsymbol{b}(\mathfrak g)$. We introduce an essentially different construction related to symmetric decompositions $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$. Poissoncommutative subalgebras $\mathcal Z,\tilde{\mathcal Z}\subset S(\mathfrak g)^{\mathfrak g_0}$ of the maximal possible transcendence degree are presented. If the $\mathbb Z_2$contraction $\mathfrak g_0\ltimes\mathfrak g_1^{\sf ab}$ has a polynomial ring of symmetric invariants, then $\tilde{\mathcal Z}$ is a polynomial maximal Poissoncommutative subalgebra of $S(\mathfrak g)^{\mathfrak g_0}$, and its free generators are explicitly described.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.00350
 Bibcode:
 2018arXiv180900350P
 Keywords:

 Mathematics  Representation Theory;
 17B63;
 14L30;
 17B08;
 17B20;
 22E46
 EPrint:
 34 pages