Poisson结构和位势
Poisson Structures and Potentials
摘要：
本文引入了势正簇上弱对数正则Poisson结构的概念。这样的泊松结构是对数正则项占主导地位的潜力。对于弱对数标准Poisson簇的相容实形式，我们指定一个实凸多面锥（簇的热带化）和紧环面的乘积上的可积系统.我们把这个理论应用到单连通半单复李群$G$的对偶Poisson-李群$G^*$上。我们在$G^*$上定义了正结构和正势，并证明了$G^*$上的自然Poisson-Lie结构相对于这个正结构和正势是弱对数正则的.对于紧实型$K \subset G$，证明了实形式$K^* \subset G^*$是相容的，并证明了相应的可积系统定义在装饰弦锥与维数为$\frac{1}{2}({\rm dim} \, G - {\rm rank} \, G)$的紧致环面的乘积上.
英文摘要：
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 $G^*$ 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 $K \subset G$ the compact real form, we show that the real form $K^* \subset G^*$ 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 $\frac{1}{2}({\rm dim} \, G - {\rm rank} \, G)$.
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