  In this paper, we introduce a study of prolongations of representations of
Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a
finite-dimensional Lie group and TG is the tangent bundle of G, by using (not
necessarily faithful) representations of G. We show that tangent functions of
Lie group actions correspond to prolonged representations. We prove that if two
representations are equivalent, then their prolongations are equivalent too. We
show that if U is an invariant subspace for a representation, then TU is an
invariant subspace for the prolongation of the given representation and vice
versa. We prove that if the prolongation of \Phi is an irreducible
representation, then \Phi is also an irreducible representation. Finally we
show that prolongations of direct sum of two representations are direct sum of
their prolongations.
