Complexifying Lie algebras (footnote to group thread)

In summary, the complexification of a real vector space V can be turned into a vector space over the complex numbers in a natural way by writing the ordered pairs (u,v) as the formal expression "u + iv" and defining addition and multiplication accordingly. This extends to a Lie algebra over the complex numbers, which has the same properties as the original real Lie algebra V. This property is known as the universal arrow property of complexification, and it has practical applications in representation theory, particularly in the case of su(2) and sl(2,C). Additionally, the categorical approach to understanding group representations can provide insights into the meaning behind them.
  • #1
marcus
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C will stand for the complexnumbers

Let V be a vectorspace over the reals.
V x V is the cartesian product, the set of ordered pairs (u,v)
We can turn that into a vectorspace over the complexnumbers
in what I guess is an obvious or at least very natural way which I'll spell out just for definiteness.

[the reason for this is that at one point in the group thread you need to complexify a Lie algebra, essentially so that you can say that an eigenvalue exists]

Turn V x V into a vectorspace VC over C
by writing the pair (u, v) as the formal expression "u + iv"
(1) add such things in the obvious way
(2) multiply them by reals in the obvious way
(3) i times "u + iv" is equal to "-v + iu"
------------------------

Now this is maybe moderately cool. Suppose V is not just a real vectorspace but is also a Lie algebra over the reals. That is, it has a bracket.

Then VC is a Lie algebra over the complexnumbers, once you define the bracket and check Jacobi.

In the spirit of the group thread, where we spell details out sometimes, and do a certain amount of routine checking, I will
spell out the VC bracket, where X1, X2, Y1, Y2 are elements of the original real Lie algebra V.

[X1 + iX2, Y1 + iY2]C = ([X1, Y1] - [X2, Y2]) + i([X1, Y2] + [X2, Y1])

This definition of the new bracket is something you don't need to memorize or accept on faith, you can work it out by the linearity (mainly the additivity) of the bracket operation. Like applying additivity once gets you
[X1 + iX2, Y1 + iY2] = [X1, Y1 + iY2] + i[X2, Y1+ iY2]
and you apply it a couple of more times and group terms.

Maybe Hurkyl already did this, the group thread is long enough now that I don't always remember.
The new Lie bracket has to be (complex) linear in each component---how we defined it really---and it has to be skew-symmetric. To see the skew-symmetry, imagine interchaning the roles of X and Y in the defintion. Everything gets multiplied by -1.

The only thing remaining to check is Jacobi
J(X,Y,Z) = [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y]] = 0

Now this is true for X,Y,Z in the original real Lie algebra V.
And keeping Y and Z fixed, J is complex linear in X.
So it is true for Y and Z in V and X = X1 + iX2.
J(X1 + iX2, Y, Z) = J(X1, Y, Z) + iJ(X2, Y, Z) = 0
Each of those terms is complex linear in Y so we
can extend this to Y = Y1 + Y2, and to Z in the same way.
The upshot is J(X,Y,Z) = 0 for X,Y,Z complex.
If in doubt, write it out.
----------------------------------

So the complex vectorspace VC actually has a bracket and is a Lie algebra.

Now this is something with a "Categorical" feel----I think Lethe would like it, maybe others too. Maybe I like it. Complexification has a "Universal" arrow property that goes like this (parroting Hall page 52)

Any complex LA, say W, can be TREATED as a real LA, by only doing scalar multiplication with real numbers!

If V is a real LA and W is a complex LA one can say what a real LA homomorphism from V into W is. You just have to check scalar multiplication is preserved using reals!

Now say we have such a φ: V --> W, a LA homomorphism from a real one into a complex one. Then φ has a UNIQUE EXTENSION to a complex homomorphism VC --> W

So the original φ is uniquely factored into a natural inclusion map V --> VC followed by the complex homomorphism extension.

φ: V --> VC --> W

Category theory has a certain air of frivolity and so it is a refreshing surprise to find that this universal property of complexification applies effectively to serious matters like the representation theory of the Lie algebra su(2). First observe that although su(2) is 2x2 matrices of complex numbers it can be treated as a real Lie algebra.

The (perhaps beautiful) fact here is that the complexification of su(2) is sl(2, C) ! Oh joy. Excuse the outburst.

And a LA representation is after all just special kind of homomorphism! Ye gods and little fishes. So any representation of su(2) EXTENDS UNIQUELY, by this universal property, to a representation of sl(2, C).

Now it just happens that the representations that are easy to discover and catalog and list and put into a simple pattern are those of sl(2, C).
Not those of the group SU(2), and not those of that groups Lie algebra su(2). Oh no, it is not what you expect! The ones we go hunting are the LA reps of the complexification sl(2, C).
 
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  • #2
Now this is something with a "Categorical" feel

Wanna know something interesting?

The "meaning" of group representations finally clicked when I was reading a page by Baez explaining it from a categorical point of view. :smile:

I was thinking of mixing in some of the category theory terminology for that reason, but I thought that it might be a little too much to swallow at once (especially since I don't know any of the cool results from CT to apply, it would be solely for the sake of the CT words)
 
  • #3
Originally posted by Hurkyl
Wanna know something interesting?

The "meaning" of group representations finally clicked when I was reading a page by Baez explaining it from a categorical point of view. :smile:

I was thinking of mixing in some of the category theory terminology for that reason, but I thought that it might be a little too much to swallow at once (especially since I don't know any of the cool results from CT to apply, it would be solely for the sake of the CT words)

Hi Hurkyl, hope I did not say too much just now in the group thread. It's your turn to prove Hall/s theorem 5.9 page 76
any two irred reps of sl(2,C) of the same dimension are equivalent. It is a fine theorem, go for it. Then we will
have classified the mothers.
 

What are Complexifying Lie algebras?

Complexifying Lie algebras are a mathematical concept used to extend a real Lie algebra to a complex Lie algebra. A complex Lie algebra is a vector space over the complex numbers equipped with a bilinear operation called the Lie bracket, which satisfies certain properties. Complexifying a Lie algebra allows for the use of complex numbers in the Lie algebra, which can be useful in certain applications.

What is the process of complexifying a Lie algebra?

To complexify a Lie algebra, one must first define a complexification map which assigns a complex number to each element in the real Lie algebra. This map must preserve the Lie bracket operation. Then, the complexification of the Lie algebra is simply the vector space over the complex numbers with the complexification map as the Lie bracket operation.

Why are complexifying Lie algebras useful?

Complexifying Lie algebras can be useful in certain applications, such as in quantum mechanics and representation theory. In these fields, complex numbers are often used to represent physical quantities, and by complexifying the Lie algebra, one can better model these systems. Additionally, complex Lie algebras have more structure and can provide more information about the underlying system.

What are some properties of complex Lie algebras?

Complex Lie algebras have many of the same properties as real Lie algebras, such as being closed under the Lie bracket operation and satisfying the Jacobi identity. However, they also have some additional properties, such as being semisimple and having a unique maximal toral subalgebra. These properties can be useful in studying and classifying complex Lie algebras.

How are complex Lie algebras related to group theory?

Complex Lie algebras are closely related to group theory, as they are often used to study and classify Lie groups. Lie groups are differentiable manifolds with a group structure, and their corresponding Lie algebras can be complexified to better understand their structure. Additionally, the representation theory of Lie groups is closely related to the representation theory of complex Lie algebras.

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