  In this paper, we present a study on the prolongations of representations of
Lie algebras. We show that a tangent bundle of a given Lie algebra attains a
Lie algebra structure. Then, we prove that this tangent bundle is algebraically
isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these,
we define prolongations of representations of Lie algebras. We show that if a
Lie algebra representation corresponds to a Lie group representation, then
prolongation of Lie algebra representation corresponds to the prolonged Lie
group representation.
