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Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces

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 نشر من قبل Jiang-Hua Lu
 تاريخ النشر 2019
  مجال البحث
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Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${mathcal{A}}_{rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${mathcal{A}}_{rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q, pi_{G/Q}, {mathcal{A}}_{rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T subset G$ is a maximal torus, $B subset G$ a Borel subgroup, and $N$ the uniradical of $B$.



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