超交换空间的李代数刻划
Lie algebaic characterization of supercommutative space
摘要：
在过去的几十年中，空间的代数化被证明是数学和理论物理之间的一个有前途的工具。从Gel'fand-Kolmogoroff和Gel'fand-Naimark的工作开始，这个分支从40年代开始向两个方向发展：一方面是通常几何空间的代数表征，另一方面是代数定义的非对易空间，这与量子引力和超弦理论密切相关。本文将这两个方面结合起来，证明了Shanks和Pursell关于紧支撑向量场的李代数的任何同构都是由基础流形的一个同构实现的经典结果的一个推广.给出了超分次空间的超李代数刻画，并明确刻画了超向量场的超李代数的同构。
英文摘要：
During the last decades algebraization of space turned out to be a promising tool at the interface between Mathematics and Theoretical Physics. Starting with works by Gel'fand-Kolmogoroff and Gel'fand-Naimark, this branch developed as from the fortieth in two directions: algebraic characterization of usual geometric space on the one hand, and algebraically defined noncommutative space, which is known to be tightly related with e.g. quantum gravity and super string theory, on the other hand. In this note, we combine both aspects, prove a superversion of Shanks and Pursell's classical result stating that any isomorphism of the Lie algebras of compactly supported vector fields is implemented by a diffeomorphism of underlying manifolds. We thus provide a super Lie algebraic characterization of super and graded spaces and describe explicitly isomorphisms of the super Lie algebras of super vector fields.
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