Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras

Pages: 141 – 165

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a9


Zhuo Chen (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Zhangju Liu (School of Mathematical Sciences, Peking University, Beijing, China)

Maosong Xiang (Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, China)


Motivated by Kapranov’s discovery of an $\operatorname{sh}$ Lie algebra structure on the tangent complex of a Kähler manifold and Chen–Stiénon–Xu’s construction of $\operatorname{sh}$ Leibniz algebras associated with a Lie pair, we find a general method to construct $\operatorname{sh}$ Leibniz algebras. Let $\mathscr{A}$ be a commutative $\operatorname{dg}$ algebra. Given a derivation of $\mathscr{A}$ valued in a $\operatorname{dg}$ module $\Omega$, we show that there exist $\operatorname{sh}$ Leibniz algebra structures on the dual module of $\Omega$. Moreover, we prove that this process establishes a functor from the category of $\operatorname{dg}$ module valued derivations to the category of $\operatorname{sh}$Leibniz algebras over $\mathscr{A}$.


$\operatorname{sh}$ Leibniz algebra, Atiyah class, commutative $\operatorname{dg}$ algebra

2010 Mathematics Subject Classification

16E45, 18G55

The first author was supported by NSFC grant 11471179.

The third author was supported by the Fundamental Research Funds for the Central Universities 3004011126.

Received 17 October 2018

Received revised 11 June 2019

Accepted 10 July 2019

Published 6 November 2019