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

Abstract

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 Poisson-commutative subalgebra, then $\mathrm{trdeg}\mathcal{C}\le \mathbf{b}(\mathfrak{g})$, where $\mathbf{b}(\mathfrak{g})=(\dim \mathfrak{g}+\mathrm{rk}\mathfrak{g})/2$. We present a method for constructing the Poisson-commutative subalgebra ${\mathcal{Z}}_{⟨\mathfrak{h},\mathfrak{r}⟩}$ of transcendence degree $\mathbf{b}(\mathfrak{g})$ via a vector space decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{r}$ into a sum of two spherical subalgebras. There are some natural examples, where the algebra ${\mathcal{Z}}_{⟨\mathfrak{h},\mathfrak{r}⟩}$ appears to be polynomial. The most interesting case is related to the pair $(\mathfrak{b},{\mathfrak{u}}_{-})$, where $\mathfrak{b}$ is a Borel subalgebra of $\mathfrak{g}$. Here we prove that ${\mathcal{Z}}_{⟨b,u_{-}⟩}$ is maximal Poisson-commutative and is complete on every regular coadjoint orbit in ${\mathfrak{g}}^{\ast }$. Other series of examples are related to decompositions associated with involutions of $\mathfrak{g}$.

23 pages