Courant algebroids, derived brackets and even symplectic supermanifolds
Courant algebroids, derived brackets and even symplectic supermanifolds
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发表时间：
1999-10
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通讯作者：
Dmitry Roytenberg
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作者：
Dmitry Roytenberg
在这篇论文中，我们研究了Courant代数胚，它最早出现在T。柯朗对狄拉克结构的研究他们后来研究了刘，温斯坦和徐谁使用柯朗代数胚推广的概念Drinfeld双李双代数胚。作为理解柯朗代数体的复杂性质的第一步，我们解释他们通过关联到每个柯朗代数体的强同伦李代数在一个自然的方式。接下来，我们提出了另一种结构的双重李双代数胚作为一个同调的哈密顿向量场的偶辛超流形。经典的BRST复形和Weil代数作为特例出现。通过导出的括号构造，我们恢复了Courant代数胚，并给出了Liu，Weinstein和Xu的加倍定理的一个简单证明。我们还介绍了一个推广，准李双代数，类似于德林费尔德的准李双代数，我们表明，在这种情况下，派生括号建设也产生了柯朗代数。最后，我们计算了S^2上单参数SU（2）-协变泊松结构族的泊松上同调。作为一个应用程序，我们表明，这些结构是非平凡的变形对方，他们不承认重新缩放。
In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complicated properties of Courant algebroids, we interpret them by associating to each Courant algebroid a strongly homotopy Lie algebra in a natural way. Next, we propose an alternative construction of the double of a Lie bialgebroid as a homological hamiltonian vector field on an even symplectic supermanifold. The classical BRST complex and the Weil algebra arise as special cases. We recover the Courant algebroid via the derived bracket construction and give a simple proof of the doubling theorem of Liu, Weinstein and Xu. We also introduce a generalization, quasi-Lie bialgebroids, analogous to Drinfeld's quasi-Lie bialgebras; we show that the derived bracket construction in this case also yields a Courant algebroid. Finally, we compute the Poisson cohomology of a one-parameter family of SU(2)- covariant Poisson structures on S^2. As an application, we show that these structures are non-trivial deformations of each other, and that they do not admit rescaling.