On the classification of semisimple Lie algebras

Abstract

Lie algebra over a field F (F= [; lis a vector space Lover F equipped with a skew symmetric bilinear operation called the Lie bracket, which satisfied the Jacobi identity. Lie algebras, have Jordan decomposition into semisimple and nilpotent parts, with representation theory of nilpotent Lie algebras being intractable in general. The finite dimensional representation and classification of semisimple Lie algebras are completely understood, after work ofElie Cartan. A classification of semisimple Lie algebras L is analyzed by choosing a Cartan subalgebra which is essentially a generic maximal subalgebra H of L on which the Lie bracket is zero (abelian). The representation of L is decomposed into weight spaces which are eigen spaces for the action of H (root space decomposition). From which the analysis of representation is easily understood by the possible weights which can occur (root system). The classification of semisimple Lie algebras by Dynkin gives the four classical Lie algebras and five exceptional simple Lie algebras over the finite algebraically field of characteristic zero.