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


Dadi Ni (School of Mathematics and Statistics, Henan University, Kaifeng, China)

Jiahao Cheng (College of Mathematics and Information Science, Center for Mathematical Sciences, Nanchang Hangkong University, Nanchang, China)

Zhuo Chen (Department of Mathematics, Tsinghua University, Beijing, China)

Chen He (School of Mathematics and Physics, North China Electric Power University, Beijing, China)


$\def\DerL{\operatorname{Der}(L)}$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 $\Gamma (\wedge^\bullet A^\vee \otimes L/A)$.


$L_infty$ 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