# Homework Help: Exercises on Lie-algebras

1. Nov 3, 2007

### CompuChip

[SOLVED] Exercises on Lie-algebras

I'm doing some homework on Lie-algebra's, and there are several things I am missing. Let me list them below in one thread.
1. The statement is "If L is a classical linear Lie-algebra, then L is semisimple." I have reduced this to proving Rad L $\subseteq$ Z(L) -- then I have that Rad L = Z(L), so L is reductive and I can use an earlier result to conclude that Rad L = 0 so L is semisimple. But I don't know how to show this. Since what I need to prove does not hold in general, I should plug in some (common) property of classical linear Lie-algebras, but I can't really think of anything useful here... any ideas?
2. Finally, I am required to prove that If any Lie-algebra L satisfies Rad L = Z(L), then all finite-dimensional representations of L in which Z(L) is represented by semisimple endomorphisms are completely reducible (Humphreys' book on Introduction to Lie-algebras and Representation theory, exercise 6.5d). So I took such a representation and found that I can simultaneously diagonalize all elements in the center, but I don't really see how this will give me any information on the reducibility of the entire representation.

I know that I haven't really posted a lot of work, but I did try to reduce the problem to the points I'm really stuck at. I obviously don't expect a fully worked answer, but I'd really appreciate some hints or general steps to solve these problems. Thanks you very much!

Last edited: Nov 4, 2007
2. Nov 4, 2007

### CompuChip

I solved and removed one of the problems. The other two remain, so any responses are welcome!

3. Nov 4, 2007

### matt grime

What is your definition of 'classical'?

4. Nov 5, 2007

### CompuChip

They are $$\mathfrak{sl}(\ell, \mathsf{F}}, \mathfrak{sp}(2\ell, \mathsf{F}}$$ and $$\mathfrak{o}(\ell, \mathsf{F}}$$ for a field $\mathsf{F}$, that is:
• $$\mathsf{A}_\ell$$, the special linear algebra of traceless endomorphisms of an (l+1)-dimensional vector space
• $$\mathsf{B}_\ell$$, the orthogonal algebra of all endomorphisms x of a (2l + 1)-dimensional vector space which satisfy f(x(v), w) = -f(v, x(w)) for the bilinear form f whose matrix is $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & I_\ell \\ 0 & I_\ell & 0 \end{pmatrix}$$.
• $$\mathsf{C}_\ell$$, the symplectic algebra of all endomorphisms x of a (2l)-dimensional vector space which satisfy f(x(v), w) = -f(v, x(w)) for the bilinear form f whose matrix is $$\begin{pmatrix} 0 & I_\ell \\ -I_\ell & 0 \end{pmatrix}$$.
• $$\mathsf{D}_{\ell \ge 2}$$, the orthogonal algebra which is constructed similarly as $$\mathsf{B}_\ell$$ but with $$s = \begin{pmatrix} 0 & I_\ell \\ I_\ell & 0 \end{pmatrix}$$.

Actually, I don't see any common property of these algebra's, except that we had to prove once that they all satisfy L = [L, L] so I thought I'd use that. But that doesn't really help (e.g. I tried proving that [Rad L, L] = 0 using the Jacobi identity but that just got me 0 = 0. I also tried dividing out something, like Rad L or Z(L) or maybe even Z(L) from Rad(L), but that didn't give anything useful either).

Last edited: Nov 5, 2007
5. Nov 5, 2007

### CompuChip

OK, so I have solved #1 as well, basically by looking at the proof in a set of course notes online, and I must admit it is non-trivial (I would never have thought of it myself, and I wasn't even close.)

As for the remaining problem, I have reduced it to showing that the
$$V_\lambda = \{ v \in V | \phi(z) v = \lambda(z) v \}$$
is completely reducible, where vector space V is the module, phi is the representation as in the question, and lambda runs over the linear forms on Z(L).

Woot, 400th post. That's fast.

OK, this was as good as trivial. Of course, the eigenspaces V_\lambda are all one-dimensional, so the proof is as good as finished.
Thanks for responding anyway matt.
Still in time for the deadline tomorrow
[/edit]

Last edited: Nov 5, 2007