On non-abelian extensions of 3-Leibniz algebras

In this paper,we study non-abelian extensions of 3-Leibniz algebras through Maurer-Cartan elements.We construct a differential graded Lie algebra and prove that there is a one-to-one correspondence between the isomorphism classes of non-abelian extensions in 3-Leibniz algebras and the equivalence classes of Maurer-Cartan elements in this differential graded Lie algebra.And also the Leibniz algebra structure on the space of fundamental elements of 3-Leibniz algebras is analyzed.It is proved that the non-abelian extension of 3-Leibniz algebras induce the non-abelian extensions of Leibniz algebras.

Omni-Representations of Leibniz Algebras

In this paper,first we introduce the notion of an omni-representation of a Leibniz algebra g on a vector space V as a Leibniz algebra homomorphism from g to the omni-Lie algebra gl(V)V.Then we introduce the omnicohomology theory associated to omni-representations and establish the relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.

Purely Hom-Lie bialgebras

In this paper, we first show that there is a Hom-Lie algebra structure on the set of(σ, σ)-derivations of an associative algebra. Then we construct the dual representation of a representation of a Hom-Lie algebra.We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some special Hom-Lie algebras using Hom-O-operators.

Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

In this paper,we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras.Explicitly,we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras.Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras.In particular,we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang(2001)in their study of bi-Hamiltonian structures.Finally,we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.