Internal symmetry of the $L_{⩽3}$ 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 $\Gamma ({\wedge }^{\bullet }{A}^{\vee }\otimes L/A)$ 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 $\mathrm{Der}(L)$ on the $L_{⩽3}$ algebra $\Gamma ({\wedge }^{\bullet }{A}^{\vee }\otimes L/A)$. Here $\mathrm{Der}(L)$ 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 $\mathrm{Der}(L)$-action internal symmetry of $\Gamma ({\wedge }^{\bullet }{A}^{\vee }\otimes L/A)$.

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References in corpus (1)

• Hochschild cohomology of dg manifolds associated to integrable distributions