关于Poisson群胚的结构  

On the Structure of Poisson Groupoids

作　　者：贺龙光[1] 

机构地区：[1]首都师范大学数学系,北京100037

出　　处：《数学学报（中文版）》1999年第5期803-808,共6页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金

摘　　要：令（ΓP,α，β）是Poisson群胚．如果它的每个α-纤维与β-纤维至多交于一点，则Γ在任一点x的特征分布有直和分解△（x）=△α（x）+△β（x），其中△α（x）Txα－1（u），△β（x）Txβ-1（u）且它们都是△（x）的辛子空间．由此得到辛叶Sx的辛子流形S和S，使在映射α之下，S辛微分同胚于P中辛叶Su，在映射β之下，S反辛微分同胚于P中辛叶Sv（定理4和5）．对于一般的Poisson群胚，也可得到类似的S和S，它们差一局部辛微分同胚是唯一确定的（定理6）．把以上结果用于辛群胚，还可得到一些更具体的性质（定理7及其推论）．Let (Γ P, α, β) be a Poisson groupoid. If its each α-fiber intersects each βfiber not more than one point, then for any x ∈Γ, α(x) = u, β(x) = v, the characteristic distribution of Γ has a decomposition into direct product △(x)= △α(x) + △β(x),such that △α(x) tangents to α-fiber, △β(x) tangents to β-fiber, △α(x) and △β(x) are both Symplectic subspaces of △(x). Since the complete integrabilities of △α(x) and △β(x), there exist Symplectic submanifolds S and S of Symplectic leaf S, such that α: S→α(S) Su is Symplectic diffeomorphism and β： S→β(S) Sv is anti-Symplectic diffeomorphism. Where Su and Sv are Symplectic leaves of P.For general Poisson groupoids, we can obtain S and S with similar properties.Using these results to Symplectic groupoids, we can obtain some more concrete results.

关 键 词：Poisson群胚 辛群胚 辛叶 微分流形 辛子流形 

分 类 号：O186.16[理学—数学]