Two and three dimensional lie algebras

In this thesis, 2 and 3 dimensional Lie algebras are considered. To this end, it is examined in how many different ways 2 dimensional and 3 dimensional non-isomorphic Lie algebras can be constructed. First, some preliminary concepts of algebra are introduced, and its fundamental properties are investigated. Then, the notions of an ideal of a Lie algebra, derived Lie algebra, the center of a Lie algebra and its derivation are introduced in order to obtain 2 and 3 dimensional non-isomorphic Lie algebras. In addition, symmetric bi-linear forms, necessary in the mentioned construction, are also given. It is proved by designating a symmetric matrix to a symmetric bi-linear form that symmetric matrices are diagonalizable. Having introduced the preliminary notions, it is shown that any 1 dimensional Lie algebra is abelian. Since the centers and derived Lie algebras of isomorphic Lie algebras are isomorphic, these notions are used to determine the non-isomorphic 2 dimensional and 3 dimensional Lie algebras. Finally, it is shown that a 2 dimensional non-abelian Lie algebra is uniquely determined whereas in 3 dimensions there are infinitely many non-isomorphic Lie algebras.