AI 中文总结
该研究在Lie–Rinehart代数的基代数与模的直和上构造F-流形代数，证明其通常非泊松，确定泊松理想判定条件，关联莱布尼茨子与Lie–Rinehart微分，得到内射锚定映射的刚性结果并给出实例。
AI 中文摘要
对于每个Lie–Rinehart代数，我们在其基代数与模的直和上构造一个F-流形代数。与文献[LodayVallette,命题13.3.26]中的断言相反，所得结构通常不是泊松结构。我们确定相伴对称泊松代数中正次数理想的幂何时为泊松理想，并将莱布尼茨子（Leibnizator）与Lie–Rinehart微分关联起来。对于常秩的有限射影模，莱布尼茨子的迹可恢复锚定映射，并得到内射锚定映射的刚性结果，最后给出代数与几何实例。
英文摘要
For every Lie--Rinehart algebra, we construct an $F$-manifold algebra on the direct sum of its base algebra and module. Contrary to the assertion in \cite[Proposition 13.3.26]{LodayVallette}, the resulting structure is generally not Poisson. We determine when powers of the positive-degree ideal in the associated symmetric Poisson algebra are Poisson ideals, and relate the Leibnizator to the Lie--Rinehart differential. For a finite projective module of constant rank, the trace of the Leibnizator recovers the anchor and yields a rigidity result for injective anchors. We conclude with algebraic and geometric examples.
Comments12 pages
