Hidden structures behind ambient symmetries of the Maurer-Cartan equation

TL;DR

This paper explores the hidden algebraic structures behind ambient symmetries of the Maurer-Cartan equation, revealing connections with dendriform, Zinbiel, Rota-Baxter algebras, and Eulerian idempotents.

Contribution

It uncovers new relationships between ambient symmetries of Maurer-Cartan elements and various algebraic structures, linking them through operads of rational functions.

Findings

Relation between gauge triviality and algebraic calculus

Connections to dendriform, Zinbiel, and Rota-Baxter algebras

Identification of new links with operad of rational functions

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.