Kapranov's construction of sh Leibniz algebras

TL;DR

This paper introduces a general method to construct sh Leibniz algebras from derivations of commutative dg algebras, extending previous specific cases and establishing a functorial relationship.

Contribution

It provides a new, broad construction technique for sh Leibniz algebras based on derivations, generalizing earlier work on special geometric structures.

Findings

Constructs sh Leibniz algebras from derivations of dg algebras.

Establishes a functor from derivations to sh Leibniz algebras.

Generalizes previous geometric constructions.

Abstract

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