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 \({{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)\) 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 \(\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}\) on G/Q is presented, in each of the coordinate charts of \({{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)\), as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl–Yakimov, making \((G/Q, \pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}, {{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q))\) into a Poisson–Ore variety. In addition, all coordinate functions in the Bott–Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure \(\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}\). 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.

令G为一个连通且简单连通的复杂半简单Lie群。对于均匀的集合ģ -spaces ģ / Q，我们构建了一个有限图谱\（{{\ mathcal {A}}} _ {{\ scriptscriptstyle BS}}（G / Q）\）上ģ / Q，被称为Bott-Samelson地图集，我们证明了其所有坐标函数都相对于G / Q上的Lusztig正结构为正。我们还显示了G / Q上的标准泊松结构\（\ pi _ {{\ scriptscriptstyle G} / {\ scriptscriptstyle Q}} \）在\（{{\ mathcal {A}}} _ {{\ scriptscriptstyle BS}}（G / Q）\）的每个坐标图中呈现为对称的Poisson CGL扩展（或其某些本地化）按照古德尔（ Goodearl–Yakimov）的意思，\（（G / Q，\ pi _ {{\ scriptscriptstyle G} / {\ scriptscriptstyle Q}}，{{\ mathcal {A}}} _ {{\ scriptscriptstyle BS}} （G / Q））\）转化为Poisson–Ore品种。此外，Bott-Samelson地图集中的所有坐标函数都显示出相对于Poisson结构\（\ pi _ {{\ scriptscriptstyle G} / {\ scriptscriptstyle Q}} \）具有完整的哈密顿量。G / Q的示例包括G本身，G / T，G / B和G / N，其中\（T \ subset G \）是最大圆环，\（B \ subset G \）是Borel子群，而N是B的单基。