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What is the relationship between the Maurer-Cartan equation $$ d\theta + \dfrac{1}{2}[\theta,\theta] = 0 $$ satisfied by Maurer-Cartan forms on Lie groups, or by pullbacks of Maurer-Cartan forms along a section of a tautological bundle $G \to G/H$ of $G$-homogeneous space, and the identically looking Maurer-Cartan equation of Kontsevich's deformation theory? Can the former be interpreted deformation-theoretically?

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    $\begingroup$ The Maurer-Cartan equation appears in deformation theory as an obstruction to upgrade a first-order solution to deformation problem to a second order ,i.e., something over $k [x]/(x^2)$ to something over $k[x]/(x^3)$. Analogously there are higher order Maurer-Cartan equations for higher order deformations (this leads to $L_{\infty}$ and dg-Lie stuff). On the other side, the Maurer-Cartan equation on Lie groups shows the vanishing of the curvature of a principal bundle. In other words, the Maurer-Cartan form on a Lie group is equivalent to a second-order deformation of your principal bundle. $\endgroup$ – user40276 May 17 '16 at 4:02

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