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

Abstract

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 3 algebra Γ(∧ • A ∨ ) ⊗ Γ(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 Diff(L) on the L 3 algebra Γ(∧ • A ∨ ) ⊗ Γ(L/A). Here Diff(L) is the space of 1-differentials 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 Γ(∧ • A ∨ ) ⊗ Γ(L/A), and for this reason we elect to call the Diff(L)-action internal symmetry of Γ(∧ • A ∨ ) ⊗ Γ(L/A).