Simple Lie algebra

Concept in Lie algebra mathematics In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. Over a field of characteristic 0, a direct sum of simple Lie algebras is called a semisimple Lie algebra. A simple Lie group is a connected Lie group whose Lie algebra is simple. Complex simple Lie algebras A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: s l n C {\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} } , s o n C {\displaystyle {\mathfrak {so}}_{n}\mathbb {C} } , s p 2 n C {\displaystyle {\mathfrak {sp}}_{2n}\mathbb {C} } (classical Lie algebras) or one of the five exceptional Lie algebras. To each finite-dimensional complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} , there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots, the nodes are jointed (or not jointed) by a number of lines depending on the angles between the simple roots and the arrows are put to indicate whether the roots are longer or shorter. The Dynki