Polynomial integrable systems from cluster structures

We present a general framework for constructing polynomial integrable systems on linearizations of Poisson varieties that admit log-canonical systems. Our construction is in particular applicable to Poisson varieties with compatible cluster or generalized cluster structures. As examples, we consider a standard complex semi-simple Poisson Lie group $G$ and a Borel subgroup $B$ of $G$, equipped with the Berenstein-Fomin-Zelevinsky cluster structures; the unipotent Lie subgroup $N_w$ of $G$ associated to any $w$ in the Weyl group of $G$, equipped with the cluster structure on the corresponding Schubert cell as first defined by Geiss-Leclerc-Schr\"oer when $G$ is simply-laced; and the dual Poisson Lie group ${\rm GL}(n, \mathbb C)^*$ of the standard Poisson Lie group ${\rm GL}(n, \mathbb C)$, equipped with the Gekhtman-Shapiro-Vainshtein generalized cluster structure. In each of these four cases, we show that every extended cluster in the respective cluster or generalized cluster structure gives rise to at least one polynomial integrable system with respect to the linearization of the Poisson structure at the identity element. For some of the polynomial integrable systems, we show that all their Hamiltonian flows are complete. Just as generalized minors on a complex semi-simple Lie group $G$ are used to describe certain initial extended clusters in the Berenstein-Fomin-Zelevinsky cluster structure on $G$, we introduce a special class of homogeneous polynomials, called signed generalized minors, on the Lie algebra $\mathfrak{g}$ of $G$, which are then used to describe some of the polynomial integrable systems obtained via our construction. As a further application, we use the homogeneous degrees of certain signed generalized minors to give an explicit formula for the index of the Lie algebra of $N_w$ for every $w$ in the Weyl group.