Simple Lie groups are a class of
Lie groups which play a role in Lie group theory similar to that of
simple groups in the theory of discrete groups. Essentially, simple Lie groups are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not
commutative. Together with the commutative Lie group of the real numbers,
and that of the unit complex numbers, U(1), simple Lie groups give the atomic "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 group SL(
n) of n by n matrices with determinant equal to 1 is simple for all
n > 1.