The Role of the Jacobi Identity in Solving the Maurer-Cartan Structure Equation
The Role of the Jacobi Identity in Solving the Maurer-Cartan Structure Equation
复制标题
雅可比恒等式在求解 Maurer-Cartan 结构方程中的作用
DOI：
10.2140/pjm.2016.282.487
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
O. Yudilevich
中科院分区：
文献类型：
--
作者：
O. Yudilevich
我们描述了一种解决与李代数相关的Maurer-Cartan结构方程的方法，该方法分离了Jacobi恒等式作为积分障碍的作用。我们表明，该方法自然地适用于另外两种有趣的情况：泊松结构的局部辛实现，在这种情况下，我们的方法揭示了泊松条件作为实现障碍的作用；以及与李代数相关的Maurer-Cartan结构方程，在这种情况下，我们得到了方程解的显式公式，该公式推广了李代数中众所周知的公式。
We describe a method for solving the Maurer-Cartan structure equation associated with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to integration. We show that the method naturally adapts to two other interesting situations: local symplectic realizations of Poisson structures, in which case our method sheds light on the role of the Poisson condition as an obstruction to realization; and the Maurer-Cartan structure equation associated with a Lie algebroid, in which case we obtain an explicit formula for a solution to the equation which generalizes the well known formula in the case of Lie algebras.