Simple Lie group

Common card · No. 471103

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...

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