与非交换形式同态相关的Poisson几何对称
Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms
摘要：
设G是特征为零的域k中所有系数为x的形式幂级数的群，F[G]是它的函数代数. C. Brouder和A. Frabetti通过F[G]的“去阿贝尔化”过程引入了一个非交换、非余交换的分次霍普夫代数H，即取F[G]的类似表示作为代数，但去掉了交换性约束.本文用一般的方法给出了H的四个单参数形变，它们是半经典极限为Poisson几何对称的量子群，如Poisson群或李双代数，即两个量子函数代数和两个量子泛包络代数.特别地，两个Poisson群是G的扩展，同构于原代数Poisson簇，但不同构于原代数群。这种分析很容易扩展到一个类似性质的霍普夫代数的hudge族，从而产生了一种方法，以一种自然的方式将一些经典的几何对称性（如泊松群和李双代数）与这种“广义对称性”联系起来：现在的情况就像是一般情况的一个最简单的玩具模型。
英文摘要：
Let G be the group of all formal power series starting with x with coefficients in a field k of zero characteristic (with the composition product), and let F[G] be its function algebra. C. Brouder and A. Frabetti introduced a non-commutative, non-cocommutative graded Hopf algebra H, via a direct process of ``disabelianisation'' of F[G], i.e. taking the like presentation of the latter as an algebra but dropping the commutativity constraint. In this paper we apply a general method to provide four one-parameters deformations of H, which are quantum groups whose semiclassical limits are Poisson geometrical symmetries such as Poisson groups or Lie bialgebras, namely two quantum function algebras and two quantum universal enveloping algebras. In particular the two Poisson groups are extensions of G, isomorphic as proalgebraic Poisson varieties but not as proalgebraic groups. This analysis easily extends to a hudge family of Hopf algebras of similar nature, thus yielding a method to associate to such "generalized symmetries" some classical geometrical symmetries (such as Poisson groups and Lie bialgebras) in a natural way: the present case then stands as a simplest, toy model for the general situation.
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