Internal symmetry of the $L_{\leqslant 3}$ algebra arising from a Lie pair

Abstract

$\def\DerL{\operatorname{Der}(L)}$A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical $L_{\leqslant 3}$ algebra $\Gamma (\wedge^\bullet A^\vee \otimes L/A)$ whose unary bracket is the Chevalley–Eilenberg differential arising from every Lie pair $(L,A)$. In this note, we prove that to such a Lie pair there is an associated Lie algebra action by $\operatorname{Der}(L)$ on the $L_{\leqslant 3}$ algebra $\Gamma (\wedge^\bullet A^\vee \otimes L/A)$. Here $DerL$ is the space of derivations on the Lie algebroid $L$, or infinitesimal automorphisms of $L$. The said action gives rise to a larger scope of gauge equivalences of Maurer–Cartan elements in $\Gamma (\wedge^\bullet A^\vee \otimes L/A)$, and for this reason we elect to call the $\DerL$-action internal symmetry of $\Gamma (\wedge^\bullet A^\vee \otimes L/A)$.