Courant-Dorfman代数及其上同调
Courant-Dorfman algebras and their cohomology
摘要：
我们引入一种新的代数类型，Courant-Dorfman代数。这些对于柯朗代数胚就像李-莱因哈特代数对于李代数胚，或者泊松代数对于泊松流形一样。我们的工作与任意环和模块，没有任何规则性，有限性或非退化的假设。对于每一个Courant-Dorfman代数$(\R,\E)$，我们通过显式公式以函子的方式关联一个微分分次代数$\C(\E,\R)$。我们描述了两个典型的过滤$\C(\E,\R)$，并推导出一个模拟的Cartan关系的衍生物的$\C(\E,\R)$，我们分类中心的$\E$扩展的$H^2(\E,\R)$和研究的典型的上循环$Θ\in\C^3(\E,\R)$，其类$[Θ]$阻碍重新缩放的Courant-Dorfman结构。在非退化的情况下，我们还明确描述了$\C(\E,\R)$上的泊松括号;对于有限维光滑流形上与Courant代数胚相关的Courant-Dorfman代数，我们证明了Poisson dg代数$\C(\E,\R)$同构于\cite{Roy4-GrSymp}中使用分次流形构造的Poisson dg代数。
英文摘要：
We introduce a new type of algebra, the Courant-Dorfman algebra. These are to Courant algebroids what Lie-Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant-Dorfman algebra $(\R,\E)$ we associate a differential graded algebra $\C(\E,\R)$ in a functorial way by means of explicit formulas. We describe two canonical filtrations on $\C(\E,\R)$, and derive an analogue of the Cartan relations for derivations of $\C(\E,\R)$; we classify central extensions of $\E$ in terms of $H^2(\E,\R)$ and study the canonical cocycle $Θ\in\C^3(\E,\R)$ whose class $[Θ]$ obstructs re-scalings of the Courant-Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on $\C(\E,\R)$; for Courant-Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra $\C(\E,\R)$ is isomorphic to the one constructed in \cite{Roy4-GrSymp} using graded manifolds.
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