反射群与二次Poisson代数的刚性
Reflection Groups and Rigidity of Quadratic Poisson Algebras
AI总结：
本文研究二次Poisson代数的不变量理论，证明了Shephard-Todd-Chevalley定理的Poisson版本，并揭示了多种代数族的Poisson刚性，即固定子环A^G与A同构当且仅当群G平凡。
摘要：
本文研究二次Poisson代数的不变量理论。设G为二次Poisson代数A的分次Poisson自同构有限群。当A的Poisson括号为斜对称时，证明了Shephard-Todd-Chevalley定理的Poisson版本，即固定Poisson子环A^G为斜对称当且仅当G由反射生成。对于许多其他著名的二次Poisson代数族，我们证明G包含有限甚至无反射。这种Poisson刚性结果确保相应的固定Poisson子环A^G作为Poisson代数不与A同构，除非G是平凡的。
英文摘要：
In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of the Shephard-Todd-Chevalley theorem is proved stating that the fixed Poisson subring A^G is skew-symmetric if and only if G is generated by reflections. For many other well-known families of quadratic Poisson algebras, we show that G contains limited or even no reflections. This kind of Poisson rigidity result ensures that the corresponding fixed Poisson subring A^G is not isomorphic to A as Poisson algebras unless G is trivial.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
