Poisson Structures and Potentials

Abstract

We introduce the 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 logcanonical 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* of a simply-connected semisimple complex Lie group G.

We define a positive structure and potential on G* and show that the natural Poisson-Lie structure on G* is weakly log-canonical with respect to this positive structure and potential. For K ⊂ G the compact real form, we show that the real form K* ⊂ G* 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 $\frac{1}{2}$ (dimG − rankG).