Simple Lie algebra
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.
In group theory, a simple Lie group is a connected locally compact non-abelian Lie group G which does not have nontrivial connected normal subgroups.
A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself (or equivalently, a Lie algebra of dimension 2 or more, whose only ideals are 0 and itself).
An equivalent definition of a simple Lie group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.
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