Differential geometry of Hom-lie Algebras and Hom-lie Algebroids

Abstract

In this paper, the geometry of Hom-Lie algebras and Hom-Lie algebroids are studied. In this direction, the effects of connections on Hom-Lie algebras and Hom-Lie algebroids are first investigated. By finding the relations among hom-curvature operator, torsion and  connection on Hom-Lie algebras, Bianchi-like identities for hom-curvature operator and torsion are found. Then, with the help of these  found properties, the relationships among Hom-Lie algebras, Hom-admissible algebras, Hom-Flexible algebras, Hom-Pre-Lie algebras  and Hom-Post algebras are examined. These relationships have been also carried to the set of vector fields of the manifold with Otsuki  connection. Then, conditions of being Hom-Lie-Admissible algebroid, Hom-Pre-Lie algebroid and Hom-Post-Lie algebroid for a Hom- bundle are obtained.