Simple Lie group

In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.

Together with the commutative Lie group of the real numbers,

R

{\displaystyle \mathbb {R} }

, and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "building blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n,

R

{\displaystyle \mathbb {R} }

) of n by n matrices with determinant equal to 1 is simple for all odd n > 1, when it is isomorphic to the projective special linear group.

The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification.

Definition

Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether

R

{\displaystyle \mathbb {R} }

is a simple Lie group.

The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed connected normal subgroup is either the identity or the whole group.