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The classification of Lie Algebra starts with the concept of the solvability of a Lie Algebra.
Consider the set of all Lie bracket products of elements in V; i.e., V_{1} = {[v,w]: v, w in V}. V_{1} is necessarily an ideal subalgebra of V because [[v,w],u]=[z,u] is necessarily an element of V_{1}. But V_{1} will not necessarily include all of the elements of V. It may for some Lie algebras but for others it may not.
Having formed V_{1} the process may be repeated. Thus V_{2}={[v,w]: v,w in V_{1}} and likewise for V_{3} and so on. At some point the process may produce the trivial subalgebra of the additive identity element 0 of V; i.e., for some n, V_{n}={0}. A Lie algebra such that the process described above terminates in the trivial subalgebra {0} is said to be solvable. The ideal subalgebras of a solvable Lie algebra are themselves solvable.
There can be ideal subalgebras of a Lie algebra that are solvable but the Lie algebra itself not solvable. For a Lie algebra its radical subalgebra is defined as the maximal solvable ideal subalgebra. It can be proven that for any Lie algebra such a radical subalgebra exists.
Any nontrivial Lie algebra V necessarily contains at least two ideal algebras; i.e., V itself and the trivial subalgebra {0}. If there are no other ideal subalgebras the Lie algebra is said to be simple.
A Lie algebra V is said to be abelian if its Lie bracket is symmetric; i.e., [u,v] = [v,u] for all u and v in V. An abelian subalgebra S of a Lie algebra V is one for which the Lie bracket is symmetric in S even if it is not symmetric for all of V. If a Lie algebra V contains no abelian ideal subalgebras other than the trivial one of {0} then V is said to be semisimple.
Suppose that for a Lie algebra V there are subsets A and B such that any element v of V can be represented as the sum of an element from A and an element from B; i.e., v=a+b where a is in A and b is in B. This relation is denoted as V=A+B. If furthermore the Lie bracket products of the elements of A and B are all equal to the additive identity of V, [a,b]=0 for all a in A and b in B, then V is said to be the direct sum of A and B. Such A and B are subalgebras of V.
If for a Lie algebra V there are two subalgebras A and B such that V=A+B and the Lie bracket products of the elements of A and B are all contained within A then V is said to be the semidirect sum of A and B.
The significance of the concepts defined above is that any Lie algebra is isomorphic to the semidirect sum of its radical and a semisimple subalgebra. The French mathematician Elie Cartan was able to classify all of the semisimple Lie algebras. They all fall into four classes except for five special Lie algebras.
(To be continued.)
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