Complexifying Lie algebras (footnote to group thread)

  1. marcus

    marcus 24,315
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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 dont 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 dont 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).
    Last edited: Aug 10, 2003
  2. jcsd
  3. Hurkyl

    Hurkyl 16,090
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    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)
  4. marcus

    marcus 24,315
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    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.
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