In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a quadratic Lie algebra. We prove a uniqueness statement about these Poisson brackets with a given moment map. As an application we get formulae for the linearization of the quasi-Poisson structure of the moduli space of flat connections on a punctured sphere, and thereby identify their symplectic leaves with the reduction of coadjoint orbits. Equivalently, we get linearizations for the Goldman double Poisson bracket, our definition of Poisson brackets coincides with that of Van Den Bergh in this case. This can furthermore be interpreted as giving a monoidal equivalence between Hamiltonian quasi-Poisson spaces and Hamiltonian spaces.

中文翻译：

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康采维奇《谎言世界》中的泊松括号

在这篇笔记中，康采维奇的“谎言世界”中泊松括号的概念得到了发展。这些括号可以被认为是“普遍”定义的经典泊松结构，即仅涉及二次李代数的结构映射的形式表达式。我们用给定的矩图证明了关于这些泊松括号的唯一性陈述。作为一个应用，我们得到了穿孔球面上平面连接模空间的拟泊松结构的线性化公式，从而通过减少共伴随轨道确定它们的辛叶。等效地，我们得到了 Goldman 双泊松括号的线性化，在这种情况下，我们对泊松括号的定义与 Van Den Bergh 的定义一致。