Poisson cohomology and quantization.
Poisson cohomology and quantization.
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DOI：
10.1515/crll.1990.408.57
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发表时间：
2013-03
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通讯作者：
J. Huebschmann
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作者：
J. Huebschmann
设R是交换环，A是R上的Poisson代数.本文在A的K ahler微分的A-模和Poisson括号上构造了一个Rinehart意义下的（R，A）-李代数结构.这就产生了适当的代数概念的泊松同调和上同调的任意泊松代数。其几何版本包括由Brylinski，Koszul和Lichnerowicz引入的泊松流形的规范同调和泊松上同调，并通过将它们分别表示为Tor和Ext群，在适当的微分算子代数上，将后者在标准同调代数中表示。此外，泊松结构决定了复计算泊松上同调的一个封闭的2-形式。这个2-形式推广了定义光滑流形上辛结构的2-形式;此外，2-形式在泊松上同调中的类推广了光滑流形上辛结构的德拉姆上同调中的类，并且似乎是构造A的合适线性表示的关键成分，被视为李代数;这种表示出现在量子理论中。为了描述这个类和构造的表示，我们涉及的连接和曲率的形式概念，推广经典的扩展李代数。我们说明了我们的结果与泊松代数的一些例子，并与一个相对论性粒子的量子化过程与零静止质量和自旋为零。
Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of K\"ahler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for an arbitrary Poisson algebra. A geometric version thereof includes the canonical homology and Poisson cohomology of a Poisson manifold introduced by Brylinski, Koszul, and Lichnerowicz, and absorbes the latter in standard homological algebra by expressing them as Tor and Ext groups, respectively, over a suitable algebra of differential operators. Furthermore, the Poisson structure determines a closed 2-form in the complex computing Poisson cohomology. This 2-form generalizes the 2-form defining a symplectic structure on a smooth manifold; moreover, the class of that 2-form in Poisson cohomology generalizes the class in de Rham cohomology of a symplectic structure on a smooth manifold and appears as a crucial ingredient for the construction of suitable linear representations of A, viewed as a Lie algebra; representations of this kind occur in quantum theory. To describe this class and to construct the representations, we relate formal concepts of connection and curvature generalizing the classical ones with extensions of Lie algebras. We illustrate our results with a number of examples of Poisson algebras and with a quantization procedure for a relativistic particle with zero rest mass and spin zero.