标题： 霍普夫作用于泊松代数 显示英文标题

标题： Hopf actions on Poisson algebras

摘要： 我们研究了有限维Hopf代数作用于Poisson代数的现象，并探讨了在这种背景下量子刚性的表现。 我们的主要关注点是过滤的（特别是二次的）Poisson代数，包括$2n$变量中的Weyl Poisson代数以及某些两个变量的Poisson代数。 特别是，我们证明了任何有限维Hopf代数如果内忠实作用于这些Poisson代数，则必须通过一个群代数分解——这与关联情况下Weyl代数的已知刚性定理相呼应。 证明的关键在于将Hopf作用提升到相关的Rees代数上，在那里我们构造合适的非交换“量子化”，从而能够利用Hopf作用于量子（或过滤）代数的分类结果。 我们还讨论了Poisson代数上的群作用如何扩展到泛包络代数上，并给出了Taft代数在某些低维Poisson代数上的部分分类。 显示英文摘要

摘要： We study finite-dimensional Hopf actions on Poisson algebras and explore the phenomenon of quantum rigidity in this context. Our main focus is on filtered (and especially quadratic) Poisson algebras, including the Weyl Poisson algebra in $2n$ variables and certain Poisson algebras in two variables. In particular, we show that any finite-dimensional Hopf algebra acting inner faithfully on these Poisson algebras must necessarily factor through a group algebra-mirroring well-known rigidity theorems for Weyl algebras in the associative setting. The proofs hinge on lifting the Hopf actions to associated Rees algebras, where we construct suitable noncommutative "quantizations" that allow us to leverage classification results for Hopf actions on quantum (or filtered) algebras. We also discuss how group actions on Poisson algebras extend to universal enveloping algebras, and we give partial classifications of Taft algebra actions on certain low-dimensional Poisson algebras.