Let $\mathcal{S}(\mathfrak{g})$ be the symmetric algebra of a reductive Lie algebra $\mathfrak{g}$ equipped with the standard Poisson structure. If $\mathcal{C}\subset \mathcal{S}(\mathfrak{g})$ is a Poisson-commutative subalgebra, then $\mathrm{tr}.\mathrm{deg}\mathcal{C}⩽\mathit{b}(\mathfrak{g})$, where $\mathit{b}(\mathfrak{g})=(dim\mathfrak{g}+\mathsf{rk}\mathfrak{g})/2$. We present a method for constructing the Poisson-commutative subalgebra ${\mathcal{Z}}_{⟨\mathfrak{h},\mathfrak{r}⟩}$ of transcendence degree $\mathit{b}(\mathfrak{g})$ via a vector space decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{r}$ into a sum of two spherical subalgebras. There are some natural examples, where the algebra ${\mathcal{Z}}_{⟨\mathfrak{h},\mathfrak{r}⟩}$ appears to be polynomial. The most interesting case is related to the pair $(\mathfrak{b},{\mathfrak{u}}_{-})$, where $\mathfrak{b}$ is a Borel subalgebra of $\mathfrak{g}$. Here we prove that ${\mathcal{Z}}_{⟨\mathfrak{b},{\mathfrak{u}}_{-}⟩}$ is maximal Poisson-commutative and is complete on every regular coadjoint orbit in ${\mathfrak{g}}^{\ast }$. Other series of examples are related to involutions of $\mathfrak{g}$.

中文翻译：

---

与 S(g) 的 2-分裂和泊松交换子代数相关的兼容泊松括号

让 $\mathcal{秒}(\mathfrak{G})$ 是还原李代数的对称代数 $\mathfrak{G}$配备标准泊松结构。如果$\mathcal{C}\subset \mathcal{秒}(\mathfrak{G})$ 是泊松交换子代数，那么 $\mathrm{tr}.\mathrm{度数}\mathcal{C}⩽\mathit{乙}(\mathfrak{G})$， 在哪里 $\mathit{乙}(\mathfrak{G})=(暗淡\mathfrak{G}+\mathsf{克}\mathfrak{G})/2$. 我们提出了一种构造泊松交换子代数的方法${\mathcal{Z}}_{⟨\mathfrak{H},\mathfrak{r}⟩}$ 超越度 $\mathit{乙}(\mathfrak{G})$ 通过向量空间分解 $\mathfrak{G}=\mathfrak{H}\oplus \mathfrak{r}$成两个球面子代数的和。有一些自然的例子，其中代数${\mathcal{Z}}_{⟨\mathfrak{H},\mathfrak{r}⟩}$似乎是多项式。最有趣的案例与这对有关$(\mathfrak{乙},{\mathfrak{你}}_{-})$， 在哪里 $\mathfrak{乙}$ 是一个 Borel 子代数 $\mathfrak{G}$. 这里我们证明${\mathcal{Z}}_{⟨\mathfrak{乙},{\mathfrak{你}}_{-}⟩}$ 是最大泊松交换的，并且在每个规则的共伴轨道上是完备的 ${\mathfrak{G}}^{＊}$. 其他系列的例子与对数有关$\mathfrak{G}$.