### Abstract:

Given a Lie algebra g and its complexi_cation gC; the representations of gC are isomorphic
to those of g. Moreover, if g is the corresponding Lie algebra of a connected
and simply connected Lie group G then the representations of the Lie group in question
are isomorphic to those of gC. This thesis explains the basic concepts of Lie
groups and Lie algebras. Further, the basic representation theory of Lie groups and
Lie algebras, particularly those of semisimple Lie algebras is discussed. In addition,
an exposition of a method of constructing induced representations, with the particular
case of the Poincaré group and an application in Physics is given. Finally, some
physical applications of Lie groups and Lie algebras are outlined and discussed.