Poisson Structures and Potentials

Abstract

We introduce a notion of weakly log-canonical Poisson structures on positive varieties with potentials. Such a Poisson structure is log-canonical up to terms dominated by the potential. To a compatible real form of a weakly log-canonical Poisson variety we assign an integrable system on the product of a certain real convex polyhedral cone (the tropicalization of the variety) and a compact torus. We apply this theory to the dual Poisson-Lie group $G^{\ast }$ of a simply-connected semisimple complex Lie group $G$. We define a positive structure and potential on $G^{\ast }$ and show that the natural Poisson-Lie structure on $G^{\ast }$ is weakly log-canonical with respect to this positive structure and potential. For $K\subset G$ the compact real form, we show that the real form $K^{\ast }\subset G^{\ast }$ is compatible and prove that the corresponding integrable system is defined on the product of the decorated string cone and the compact torus of dimension $\frac{1}{2}(\dim G-\mathrm{rank}G)$.

31 pages