Derived brackets for fat Leibniz algebras

Cai, Xiongwei; Liu, Zhangju

doi:10.1016/j.geomphys.2019.103524

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“…By some similar computations (see [5]), one can easily show that (Dω) (0) k also satisfies the weak symmetry property (3.1). This completes the proof.…”

Section: Cohomology Of Hemistrict Lie 2-algebrasmentioning

confidence: 78%

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Cohomology of hemistrict Lie 2-algebras

Cai

Liu

Xiang

2020

Communications in Algebra

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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 ...

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“…By some similar computations (see [5]), one can easily show that (Dω) (0) k also satisfies the weak symmetry property (3.1). This completes the proof.…”

Section: Cohomology Of Hemistrict Lie 2-algebrasmentioning

confidence: 78%

“…The cohomology of the representation V is defined to be the cohomology of C • (L, V ) (see Definition 3.4). Applying this construction of standard complex to Leibniz algebras, it is shown in [6] that the Leibniz bracket of a fat Leibniz algebra can be realized as a derived bracket.…”

Section: Introductionmentioning

confidence: 99%