MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The Lie algebra $so_5$ has 10 elements and its root structure is given by the Dynkin diagram B2. I have been having trouble creating an explicit $5 \times 5$ complex matrix representation of its 10 elements from its Cartan matrix. I would greatly appreciate help with this.

share|cite|improve this question

closed as no longer relevant by S. Carnahan Jan 31 '11 at 3:19

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

I am not sure I understand this question. Why would you expect the Cartan matrix to give you a particular matrix representation? The Cartan matrix gives you the structure of the Lie algebra using the Serre relations. This is explained in a variety of places, e.g., Humphrey's book on Lie algebras and representation theory. $$ $$ The compact real form of the Lie algebra of type B2 is the algebra of skewsymmetric endomorphisms of a five-dimensional euclidean space. It doesn't get any more explicit than that: just take the 5x5 skewsymmetric matrices. – José Figueroa-O'Farrill Jan 16 '11 at 20:28
The question is: what are the matrices corresponding to the Chevalley generators, i.e. the famous (e_i,f_i,h_i)? – Guntram Jan 16 '11 at 21:26
@Guntram: that's also not a research-level question, Allen's answer notwithstanding. See his answer for how to go about it. In any case, you don't get this from the Cartan matrix. – José Figueroa-O'Farrill Jan 17 '11 at 6:54
Thanks, I have found this answer helpful. Also, I have just discovered the book, "Notes on Lie Algebras" by Hans Samelson, which has clarified the issue. – garretstar Jan 17 '11 at 18:13

While I'm not sure this question is appropriate for this site, here goes.

First, you need a maximal torus. Inside SO(5) we have SO(4) and then SO(2) x SO(2).

Write your antisymmetric matrices as

$$\begin{matrix} aJ & C & v \end{matrix} $$ $$\begin{matrix} -C^T & bJ & w \end{matrix} $$ $$\begin{matrix} -v^T & -w^T & 0 \end{matrix} $$ where $J = \left( {0\atop -1}{1\atop 0} \right) $, $C$ is square, and $v$ and $w$ are columns. Then the $a$ and $b$ parts are the torus, the $v$ gets you the $\pm x_1$ weights, the $w$ gets you the $\pm x_2$, and the $C$ gets you the $\pm x_1\pm x_2$.

Taking $x_1$ and $x_2 - x_1$ as simple roots, the $e_{x_1}$ is $$ 0 0 0 0 1 $$ $$ 0 0 0 0 i $$ $$ 0 0 0 0 0 $$ $$ 0 0 0 0 0 $$ $$ -1 -i 0 0 0 $$ and the $e_{x_2-x_1}$ is $$ 0 0 1 i 0 $$ $$ 0 0 i -1 0 $$ $$ -1 -i 0 0 0 $$ $$ -i 1 0 0 0 $$ $$ 0 0 0 0 0 $$ Sorry for the ugly matrices -- I'm having trouble getting the matrix environment working here.

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.