The topic of this lecture is differential geometry on quantum groups. Several lecturers at this conference have talked about this subject. For a review and a fairly extensive list of references I would like to point the reader to the contributions of B. Jurco and S. L. Woronowicz in this proceedings. Here we shall propose a new rigid framework for the so-called Cartan calculus of Lie derivative...

We study the structure of subgroups of minimal connected simple groups of finite Morley rank. We first establish a Jordan decomposition for a large family of minimal connected simple groups including those with a non-trivial Weyl group. We then show that definable, connected, solvable subgroups of such a simple group are the semi-direct product of their unipotent part extended by a maximal toru...

we prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

For a given Jacobi-Jordan algebra A and vector space V over field k , non-abelian cohomological type object H 2 ( ) is constructed: it classifies all algebras containing as subalgebra of codimension equal to dim . Any such isomorphic so-called unified product ♮ Furthermore, we introduce the bicrossed (semi-direct, crossed, or skew crossed) ⋈ associated two special case product. Several examples...

For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semi-direct product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference...

We consider WZW models based on the non-semi-simple algebras that they were recently constructed as contractions of corresponding algebras for semi-simple groups. We give the explicit expression for the action of these models, as well as for a generalization of them, and discuss their general properties. Furthermore we consider gauged WZW models based on these non-semi-simple algebras and we sh...