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# Pure and Applied Mathematics Quarterly

## Volume 19 (2023)

### Number 4

### Special Issue in honor of Victor Guillemin

Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe

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

Pages: 2195 – 2234

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a16

#### Authors

#### 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 $\Gamma (\wedge^\bullet A^\vee \otimes L/A)$.

#### Keywords

$L_infty$ algebra, $L_{\leqslant 3}$ algebra, $\operatorname{dg}$ algebra, Lie pair, Lie algebra action

#### 2010 Mathematics Subject Classification

16E45, 17B70, 58A50

The authors’ research is supported by NSFC grant 12071241; by the Research Fund of Nanchang Hangkong University (EA202107232); and by the Fundamental Research Funds for the Central Universities (2020MS040).

Received 30 January 2022

Received revised 2 June 2022

Accepted 27 June 2022

Published 20 November 2023