关于Poisson代数的半中心
On the semi-centre of a Poisson algebra
摘要：
若$\mathfrak{g}$是李代数，则Poisson代数$S(\mathfrak{g})$的半中心是由ad $(\mathfrak{g})$ -特征向量生成的子代数.在本文中，我们抽象这个定义的背景下，积分泊松代数。给出了Poisson半中心$A^{\operatorname{sc}}$是按其权空间分次的Poisson代数的充要条件.在这种情况下，我们证明了泊松半中心具有许多很好的性质：有理Casimir是泊松正规元的等价物，泊松Dixmier-MÜglin等价对半中心成立。
英文摘要：
If $\mathfrak{g}$ is a Lie algebra then the semi-centre of the Poisson algebra $S(\mathfrak{g})$ is the subalgebra generated by ad$(\mathfrak{g})$-eigenvectors. In this paper we abstract this definition to the context of integral Poisson algebras. We identify necessary and sufficient conditions for the Poisson semi-centre $A^{\operatorname{sc}}$ to be a Poisson algebra graded by its weight spaces. In that situation we show the Poisson semi-centre exhibits many nice properties: the rational Casimirs are quotients of Poisson normal elements and the Poisson Dixmier-Mœglin equivalence holds for the semi-centre.
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