Prolongations of representations of lie groups to tangent bundles

In this study, representations defined on any finite dimensional Lie group G was prolonged to tangent bundle of G by using geometric methods. Also relations between algebraic structures of G and the tangent bundle of G were obtained. By using these structures, algebraic structures on tangent bundles were obtained and prolongations of representations of Lie algebras are defined. Using one-to-one correspondence of homogeneous vector bundles and representations, we obtain corresponding homogeneous bundles of prolonged representations.