Reductive homogeneous spaces and nonassociative algebras

These are the notes (or better, a first version of them) of a course to be given at a CIMPA research school, to be held in Marrakech (April 13-24, 2015). The title of the school is ‘Geometrie differentielle et algebres non associatives’. The purpose of the course is to present a classical result by Nomizu [Nom54] that relates the invariant affine connections on reductive homogeneous spaces to nonassociative algebras defined on the tangent space of a point. This allows us to give precise algebraic descriptions of these connections, of their torsion and curvature tensors, ... Nomizu’s result constitutes a very nice bridge between these two areas mentioned in the title of the school. Modulo basic results, the presentation is self contained. The first three sections review the results on smooth manifolds, on affine connections and on Lie groups and Lie algebras that will be needed for our purpose. The fourth section is devoted to study the invariant affine connections on homogeneous spaces, following the presentation in [AVL91, Chapter IV]. Also, Lie-Yamaguti algebras will be introduced here. Nomizu’s Theorem is presented and proved in Section 5, based on the previous work. The proof given by Nomizu in his seminal paper is quite different. Finally, in Section 6 left and bi-invariant affine connections in Lie groups will be studied. Several classes of algebras, like Lie-admissible, flexible, associative or left-symmetric, appear naturally. Several exercises, complementing the theory, are scattered through the text.