Journal of Generalized Lie Theory and Applications

The biography and bibliography of one of the great contemporary geometers Maks A. Akivis and the description of his scientific activities are presented.

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.

In this paper we study quantizations, associativity constraints and braidings in the monoidal category of monoid graded modules over a commutative ring. All of them can be described in terms of the cohomology of underlying monoid. The case when the monoid is a finite topology has the main interest for us. The cohomology classes which are invariant with respect to homeomorphism group produce remarkable algebraic constructions. We study in details the Sierpinski and discrete topology and show the relations with the Clifford algebras, the Cayley algebra and their quantizations. All of them are ®-associative and ¾-commutative for suitable associativity constraints ® and braidings ¾.

Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation are found. In particular, the Lie algebra of the multiplication group of the birepresentation is explicitly given.