Let K be an infinite field. We prove that if a variety of anticommutative K-algebras-not necessarily associative, where xx " 0 is an identity-is locally algebraically cartesian closed, then it must be a variety of Lie algebras over K. In particular, Lie K is the largest such. Thus, for a given variety of anti-commutative K-algebras, the Jacobi identity becomes equivalent to a categorical condition: it is an identity in V if and only if V is a subvariety of a locally algebraically cartesian closed variety of anti-commutative K-algebras. This is based on a result saying that an algebraically coherent variety of anti-commutative K-algebras is either a variety of Lie algebras or a variety of anti-associative algebras over K.

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we introduce crossed modules of Hom-Lie algebras showing their equivalence with cat 1 -Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

Available online xxxx Communicated by Volodymyr Mazorchuk MSC: 17B55 17B30 17B60 Keywords: Lie superalgebras Associative superalgebras Non-abelian tensor and exterior products Non-abelian homology Cyclic homology Hopf formula Crossed moduleWe introduce the non-abelian tensor product of Lie superalgebras and study some of its properties. We use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.