Higher Derived Brackets for Arbitrary Derivations

Nonabelian higher derived brackets

Let M be a graded Lie algebra, together with graded Lie subalgebras L and A such that as a graded space M is the direct sum of L and A, and A is abelian. Let D be a degree one derivation of M squaring to zero and sending L into itself, then Voronov's construction of higher derived brackets associates to D a L-infinity structure on A[-1]. It is known, and it follows from the results of this paper, that the resulting L-infinity algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras i : (L,D,[, ]) -> (M,D,[, ]). We prove this fact using homotopical tra…

Higher derived brackets and homotopy algebras

We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $Δ$. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of $Δ^2$. This allows to control higher Jacobi identities in terms of the ``order'' of $Δ^2$. Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin--Vilkovisky algebras. There is a gener…

Derived bracket construction and anti-cyclic subcomplex of Leibniz (co)homology complex

An arbitrary Leibniz algebra can be embedded in a differential graded Lie algebra via the derived bracket construction. Such an embedding is called a derived bracket representation. We will construct the universal version of the derived bracket representation, prove that the principal part of the target dg Lie algebra defines a subcomplex of Leibniz (co)homology complex and that the existence of the subcomplex is a reflection of the anti-cyclicity of the Leibniz operad.

$δ$-superderivations of KKM Double

We described $δ$-derivations and $δ$-superderivations of simple Jordan superalgebra <<KKM Double>> (also known as superalgebra of Jordan brackets) and unital simple finite-dimensional Jordan superalgebras over algebraic closed fields with characteristic $p\neq2$. As a consequence, we received relationship between non-trivial $δ$-derivations and specialty of simple superalgebra <<KKM Double>>. We constructed new examples of non-trivial 1/2-derivations of Jordan superalgebras of vector type.

n元Hom-Nambu超代数与Hom-Lie超代数的完备性

On Completeness of $n$-ary Hom-Nambu and Hom-Lie Superalgebras

本文研究乘法n元Hom-Nambu超代数的完备性，建立相关判据，给出非n元Hom-Lie超代数的完备例子，分析直和、扭曲下中心性质及低维分类中的完备成员。

Bicharacters, braids and Jacobi identity

For an abelian group G we consider braiding in a category of G-graded modules $M^{kG}$ given by a bicharacter χon G. For $(G,χ)$-bialgebra A in $M^{kG}$ an analog of Lie bracket is defined. This bracket is determined by a linear map $E\in\End(A)$ and n-ary operations $Ω^{n}_{E}$ on A. Our result states that if $E(1)=0,E^{2}=0$ and $Ω^{3}_{E}=0$ then a braided Jacobi identity holds and the linear map E is a braided derivation of a braided Lie algebra.

Lie brackets on Hopf algebra cohomology

By work of Farinati, Solberg, and Taillefer, it is known that the Hopf algebra cohomology of a quasi-triangular Hopf algebra, as a graded Lie algebra under the Gerstenhaber bracket, is abelian. Motivated by the question of whether this holds for nonquasi-triangular Hopf algebras, we show that Gerstenhaber brackets on Hopf algebra cohomology can be expressed via an arbitrary projective resolution using Volkov's homotopy liftings as generalized to some exact monoidal categories. This is a special case of our more general result that a bracket operation on cohomology is preserved under exact mono…

Derivation Algebras of Centerless Perfect Lie Algebras Are Complete

It is proved that the derivation algebra of a centerless perfect Lie algebra of arbitrary dimension over any field of arbitrary characteristic is complete and that the holomorph of a centerless perfect Lie algebra is complete if and only if its outer derivation algebra is centerless.