Semisimple Lie Group

There are three equivalent characterization of semisimple Lie algebras. The first characterization is one which is isomorphic to a direct sum of simple Lie algebras. The second characterization is that complexification of the Lie algebra of a compact simply-connected group, for example, that sl(n;C) ≅ su(n)_C is semisimple. The third characterization is that a Lie algebra g is semisimple if and only if it has the complete reducibility property, that is, if and only if every finite-dimensional representation of g decomposes as a direct sum of irreducibles. 

Recall that a group or Lie algebra is said to have the complete reducibility property if every finite-dimensional representation of it decomposes as a direct sum of irreducible invariant subspaces. A connected compact matrix Lie group always has this property. It follows that the Lie algebra of compact simply-connected matrix Lie group also has the complete reducibility property, since there is a one-to-one correspondence between the representations of the compact group and its Lie algebra. Because there is a one-to-one correspondence between the representations of a real Lie algebra and the complex-linear representations of its complexification, we see also that if a complex Lie algebra g is isomorphic to the complexification of the Lie algebra of a compact simply-connected group, then g has the complete reducibility property. We have seen this reasoning to sl(2;C) (the complexification of the Lie algebra of SU(2) and to sl(3;C) and the complexification of the Lie algebra SU(3)).

Complex semisimple Lie algebras are complex Lie algebras that are isomorphic to the complexification of the Lie algebra of a compact simply-connected matrix Lie group.

Definition

If g is a complex Lie algebra, the an ideal in g is a complex subalgebra h of g with the property that for all X in g and H in h, we have [X, H] in h. 

A complex Lie algebra g is called indecomposable if the only ideals in g are g and {0}. A complex Lie algebra g is called simple if g is indecomposable and dim ≥ 2.

A complex Lie algebra is called reductive if it is isomorphic to a direct sum of indecomposable Lie algebras. A complex Lie algebra is called semisimple if it is isomorphic to a direct sum of simple Lie algebras. Note that a reductive Lie algebra is a direct sum of indecomposable algebras, which are either simple or one-dimensional commutative. Thus, a reductive Lie algebra is one that decomposes as a direct sum of a semisimple algebra and a commutative algebra. 

The following table lists the complex Lie algebras that are either reductive (not semisimple) or semisimple.

sl(n;C) (n≥2) semisimple
so(n;C) (n≥3) semisimple 
so(2;C) reductive 
gl(n;C) (n≥1) reductive 
sp(n;C) (n≥1) semisimple

All of the above listed semisimple algebras are actually simple, except for so(4;C), which is isomorphic to sl(2;C) ⊕ sl(2;C). Every complex simple Lie algebra is isomorphic to one of sl(n;C), so(n;C) (n≠4), sp(n;C), or to one of the five "exceptional" Lie algebras conventionally called G₂, F₄, E₆, F₇, and E₈.

For real Lie algebra,

su(n) (n≥2) semisimple 
so(n) (n≥3)  semisimple 
so(2) reductive 
sp(n) (n≥1) semisimple 
so(n,k) (n+k ≥3) semisimple 
so(1,1) reductive 
sp(n;R) (n≥1) semisimple 
sl(n;R) (n≥2) semisimple
gl(n;R) (n≥1) reductive

In each case, the complexification of the listed Lie algebra is isomorphic to one of the complex Lie algebras in the above table. Note that the Heisenberg group, the Euclidean group, and the Poincare group are neither reductive nor semisimple.