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

Zhuo Chen ; Zhangju Liu ; Maosong Xiang

Homology, homotopy, and applications, Tome 22 (2020) no. 1, pp. 141-165.

Résumé

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 $\mathscr{A}$.

Détail

DOI : 10.4310/HHA.2020.v22.n1.a9

Classification : 16E45, 18G55
Keywords: $\operatorname{sh}$ Leibniz algebra, Atiyah class, commutative $\operatorname{dg}$ algebra