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 .

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The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an SH Lie algebra A when it is extended to L. In fact, given such an SH Lie pair (L, A), and any A-module E, there associates a canonical cohomology class, the Atiyah class [α E ], which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class [α L/A ] induces a graded Lie algebra structure on H • CE (A, L/A[−2]), and the Atiyah class [α E ] of any A-module E induces a Lie algebra module structure on H • CE (A, E). Moreover, Atiyah classes are invariant under gauge equivalent A-compatible infinitesimal deformations of L.

We study representations of hemistrict Lie 2-algebras and give a functorial construction of their cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra L and the cohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of L are isomorphic to the Chevalley-Eilenberg cohomology of the induced Lie algebra L Lie .[−, −] g induces a semistrict Lie 2-algebra G = (K[1] → g, l 2 , l 3 ). As a consequence, we haveTheorem (see Theorem 4.12). Let g be a Leibniz algebra with Leibniz kernel K and V a representation of the associated hemistrict Lie 2-algebra L g to g such that l α = 0 for all α ∈ K. ThenThe sequel(s). We plan to write two sequels to this paper, in which we address several issues not covered here: In the first one in preparation, we study cohomology of weak Lie 2-algebras, which encodes cohomology of hemistrict Lie 2-algebras and semistrict Lie 2-algebras into a unified framework. Our first goal is to establish the compatibility between the functor of taking cohomology and the skew-symmetrization functor [18] from the category of weak Lie 2-algebras to the category of semistrict Lie 2-algebras. Meanwhile, in their work [10] of weak Lie 2-bialgebras, Chen, Stiénon and Xu developed an odd version of big derived bracket approach to semistrict Lie 2-algebras. Our second goal is to establish a derived bracket formalism for cohomology of weak Lie 2-algebras.In the second one, we consider weak L ∞ -algebroids which encodes a Courant algebroid as a 2-term weak L ∞ -algebroid. According to Kontsevich and Soibelman [12], L ∞ -algebras correspond to formal pointed dg manifolds, while A ∞ -algebras correspond to noncommutative formal pointed dg manifolds. Our purposes are to reinterpret weak L ∞ -algebroids as certain commutative up to homotopy formal dg manifolds and to investigate their relation with shifted derived Poisson manifolds studied in [4] and strongly homotopy Leibniz algebra over a commutative dg algebra studied in [9]. Representations of hemistrict Lie 2-algebrasIn this section, we study representations of hemistrict Lie 2-algebras on 2-term cochain complexes as Beck modules of the category of hemistrict Lie 2-algebras.2.1. Dg Leibniz algebras and hemistrict Lie 2-algebras. As the first step, we collect some basic facts of dg Leibniz algebras, which are Leibniz algebras (or Loday algebras) introduced by Loday [16] in the category of cochain complexes (see [11] for more structure theorems on Leibniz algebras). Denote by K the base field which is either R or C.Definition 2.1. A dg (left) Leibniz algebra g over K is a (left) Leibniz algebra in the category of cochain complexes of K-vector spaces, i.e., a cochain complex g = (g • , d) together with a cochain map [−, −] g : g ⊗ g → g, called Leibniz bracket, satisfying the (left) Leibniz rule:Example 2.2. Let h = (h • , d, [−, −] h ) be a dg Lie algebra equipped with a representation ρ on a cochain complex V = (V • , d). Then the direct sum cochain complex h ⊕ V = (h • ⊕ V • , d), equipped with the bilinear map ...