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In the previous lecture we studied Lie groups and the Lie algebra that is associated with the Lie group.

Now in the next lecture, not today, but in the next lecture we'll go the opposite way.

We'll start with a Lie algebra. We want to study which Lie groups and which part of a Lie group.

We can actually go back to starting from the algebra, so how to reconstruct the group from the algebra.

Now today, however, we'll remain on the Lie algebra level and ask a question that's quite independent

of the question where the Lie algebra comes from.

We'll study a classification of all Lie algebras and that's possible,

but it's particularly simple if we study complex Lie algebras.

So if we require that the Lie algebra we study

consists as always of a vector space and a Lie bracket, which you may now take as an abstract Lie bracket,

but this L is a complex vector space.

Now of course, such complex Lie algebras can be generated, for instance, from a complex Lie group.

So example, if G is a complex Lie group, which simply means that the underlying smooth manifold G

is a complex manifold as we defined it when we studied various transition conditions between charts,

and you may remember that for a complex manifold, the transition functions didn't only have to be smooth,

they had to satisfy the Cauchy-Riemann equations.

Anyway, we're going to consider complex Lie algebras in this sense today,

and the key theorem that brings about the promised classification is by Levy.

Today I'm not going to prove very much, so I will give you several results without proof,

but I will be conceptually clean in presenting the entire construction of this classification

and the theorems that link together the various constructive elements,

and that's of course important for applications.

So the theorem we want to work towards is that every finite dimensional Lie algebra,

well, we said complex Lie algebra, consisting of a complex vector space L,

some abstract Lie bracket, bracket, can be decomposed as,

so the L can be written as a Lie subalgebra R, a direct sum,

Lie algebra L1, direct sum, dot dot dot, plus a finite number of such L's.

Now, this would be great, in fact it's a little bit more complicated.

This here, these here are direct sums, and I'll say something about this in a second,

and this is a so-called semi-direct sum, and I'll have to explain this.

Now, if you have a, so I'll explain what R and the L's are.

Where? A.

So let's start with R, and also what these direct sums and semi-direct sums are,

because on the level of a vector space you know what the direct sum is.

The direct sum is simply a Cartesian product to which you inherit the addition,

so the vector space operations on the various vector spaces component-wise.

We have this, and, but now we're not dealing with a mere vector space.

The vector space is equipped with more structure, namely a Lie bracket,

and then this is a direct sum between Lie algebras, and therefore we'll have to have

a compatibility condition also for this bracket, and I'll provide this for the semi-direct,

for the direct product here, and the semi-direct product also,

or particularly the semi-direct product, has a different compatibility condition

with respect to this bracket. So I'll explain all of this in a second.

So what are these R, L1, to Ln?

So this R is a Lie sub-algebra of L, and it has this particular property that it is solvable,

and the solvability condition is the following.

So you consider the bracket of R with R.

Now this is of course supposed to indicate the set of all R1, R2 commutator

where R1 and R2 both come from R.