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

Abstract

Motivated by Kapranov's discovery of an sh Lie algebra structure on the tangent complex of a Kähler manifold and Chen-Stiénon-Xu's construction of sh Leibniz algebras associated with a Lie pair, we find a general method to construct sh Leibniz algebras. Let A be a commutative dg algebra. Given a derivation of A valued in a dg module Ω, we show that there exist sh Leibniz algebra structures on the dual module of Ω. 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 A .

Dg Loday-Pirashvili modules over Lie algebras

“…In [7], Atiyah introduced a cohomology class, which has become to be known as Atiyah class, to characterize the obstruction to the existence of holomorphic connections on a holomorphic vector bundle. The notion of Atiyah classes have been extensively developed in the past decades for diverse purposes (see [9,11,13,14,16,29,30,35]). In this section, we recall the twisted Atiyah class defined in [13] and then compute such classes in the setting of dg Loday-Pirashvili modules.…”

“…The notion of Atiyah classes have been extensively developed in the past decades for diverse purposes (see [9,11,13,14,16,29,30,35]). In this section, we recall the twisted Atiyah class defined in [13] and then compute such classes in the setting of dg Loday-Pirashvili modules.…”

“…Then ∂ : A → Ω is an Ω-valued derivation of A . Examples of dg module valued derivations arise from dg Lie algebroids and Lie pairs can be found in [13].…”

“…In fact, by Proposition 2.4, a dg Loday-Pirashvili module structure on V is equivalent to an Ω g ⊗ V ∨ [−1]-valued dg derivation of Ω g . In an earlier work of the first and third named authors and Liu [13], we have introduced the notion of twisted Atiyah class associated with dg derivations of general dg algebras (see Proposition-Definition 2.8). Now of course we have to find the corresponding twisted Atiyah classes arising from dg Loday-Pirashvili modules.…”

“…All structure maps happen to come from twisted Atiyah classes. Section 3 also expands on the material in our previous work [13], namely a construction of higher algebraic objects, called Kapranov Leibniz ∞ [1]-algebras. We then apply this construction, namely the Kapranov functor, to the dg derivations arising from dg Loday-Pirashvili modules over Lie algebras.…”