Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices
Compatible Poisson brackets, quadratic Poisson algebras and classical r-matrices
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兼容的泊松括号、二次泊松代数和经典 r 矩阵
DOI：
10.1007/978-3-642-00873-3_15
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
T. Skrypnyk
中科院分区：
文献类型：
--
作者：
V. Roubtsov;T. Skrypnyk
我们表明，对于一个一般的二次泊松括号，它是可能的，以定义许多相关的线性泊松括号：线性化的初始括号在特殊点的邻域。证明了所构造的线性Poisson括号与初始二次Poisson括号总是相容的，并将所得结果应用于标准二次矩阵括号和经典的“扭曲反射代数”括号的情形.在第一种情况下，我们得到了只存在一个非等价线性化：标准线性矩阵括号，并恢复了标准二次括号与线性矩阵括号相容的著名结果，证明了经典扭反射方程代数括号存在大量的非等价线性化，它们都与初始二次括号相容。
We show that for a general quadratic Poisson bracket it is possible to define a lot of associated linear Poisson brackets: linearizations of the initial bracket in the neighborhood of special points. We prove that the constructed linear Poisson brackets are always compatible with the initial quadratic Poisson bracket.We apply the obtained results to the cases of the standard quadraticr-matrix bracket and to classical “twisted reflection algebra” brackets. In the first case we obtain that there exists only one non-equivalent linearization: the standard linearr-matrix bracket and recover well-known result that the standard quadratic and linearr-matrix brackets are compatible.We show that there are a lot of non-equivalent linearizations of the classical twisted Reflection Equation Algebra bracket and all of them are compatible with the initial quadratic bracket.