Hidden structures behind ambient symmetries of the Maurer-Cartan equation

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.

💡 Research Summary

The paper investigates two natural group actions on Maurer–Cartan (MC) elements of a differential graded Lie algebra (dg‑Lie algebra) 𝔤: the classical gauge action and the “ambient” action coming from Lie∞‑isotopies (i.e. Lie∞‑automorphisms whose arity‑one component is the identity). The authors’ original motivation was a technical lemma needed in a joint work with Vaintrob and Vallette: if an ambient isotopy λ is a degree‑zero cycle in the twisted complex 𝔟_φ and is homologous to zero, then its action on any MC element α is gauge‑equivalent to α. While this result was previously proved by a homotopy‑theoretic argument, the present note provides explicit formulas for the gauge trivialization and uncovers deep connections with several combinatorial algebraic structures.

The paper proceeds as follows. After recalling the “differential trick” (adjoining an internal differential element δ with