Hidden structures behind ambient symmetries of the Maurer-Cartan equation

Vladimir Dotsenko, Sergey Shadrin
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Abstract

For every differential graded Lie algebra $\mathfrak{g}$ one can define two different group actions on the Maurer-Cartan elements: the ubiquitous gauge action and the action of $\mathrm{Lie}_\infty$-isotopies of $\mathfrak{g}$, which we call the ambient action. In this note, we explain how the assertion of gauge triviality of a homologically trivial ambient action relates to the calculus of dendriform, Zinbiel, and Rota-Baxter algebras, and to Eulerian idempotents. In particular, we exhibit new relationships between these algebraic structures and the operad of rational functions defined by Loday.
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毛勒-卡尔坦方程环境对称性背后的隐藏结构
对于每一个微分级联代数 $\mathfrak{g}$,我们都可以在毛勒-卡尔坦元素上定义两种不同的群作用:无处不在的量规作用和 $\mathrm{Lie}_\infty$-isotopies of $\mathfrak{g}$的作用,我们称之为环境作用。在本注释中,我们将解释同源琐碎环境作用的几何琐碎性断言是如何与树枝形、津比尔和罗塔-巴克斯特代数以及欧拉幂等的计算相关联的。特别是,我们展示了这些代数结构与洛代定义的有理函数操作数之间的新关系。
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