Emzar Khmaladze

Crossed modules for Leibniz n-algebras

Abstract We introduce crossed modules for the category of Leibniz n-algebras and prove that they are equivalent to cat1-Leibniz n-algebras and internal categories in Leibniz n-algebras. We interpret the set of equivalence classes of crossed extensions as the second cohomology of Leibniz n-algebras developed in [Casas J. M., Loday J.-L., Pirashvili T.: Leibniz n-algebras. Forum Math. 14 (2002), 189–207].

CYCLIC HOMOLOGIES OF CROSSED MODULES OF ALGEBRAS

The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzickis excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence.

More on crossed modules in Lie, Leibniz, associative and diassociative algebras

Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are constructed. The well-known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.

Non-abelian tensor product and homology of Lie superalgebras

Abstract We 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 Millers theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.

NON-ABELIAN TENSOR AND EXTERIOR PRODUCTS MODULO q AND UNIVERSAL q-CENTRAL RELATIVE EXTENSION OF LIE ALGEBRAS

The notions of tensor end exterior products moduloq of two crossed P-modules, where q is a positive integer and P is a Lie algebra, are introduced and some properties are established. The condition for the existence of a universal q-central relative extension of a Lie epimorphism is given and this extension is described as an exterior product modulo q.