Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I request help with this is a question from Introduction to Lie algebra by Erdmann and Wildon.

The question asks to show that show that $so(4,\mathbf{C})\cong sl(2,\mathbf{C}) \oplus sl(2,\mathbf{C})$ by first showing that the set of diagonal matrices in $so(4,\mathbf{C})$ forms a Cartan subalgebra of $so(4,\mathbf{C})$ and determining the corresponding root space decomposition.

I have done the first part but Im having difficulty in finding the root space decomposition and using that to establish the isomorphism.

The book defines $so(4,\mathbf{C})$ to be a subalgebra of $gl(n,\mathbf{C})$ given by $$x\in gl(n,\mathbf{C}) :x^tS=-Sx$$ with $S$ taken to be the matrix with $l \times l$ blocks.

PS: this is not homework.

share|improve this question
    
Are you sure about the wording of the problem? This lie algebra consists of skew-symmetric matrices and the only such diagonal matrix is the zero matrix, so these wouldn't form a Cartan subalgebra. –  Santiago Canez Dec 11 '12 at 17:53
    
@SantiagoCanez See the addition made. Maybe, it'll help. –  sawyer Dec 11 '12 at 18:19

1 Answer 1

I will assume you mean $S = \begin{pmatrix} 0&I_2\\I_2&0 \end{pmatrix}$. If so, please edit your question accordingly.

You should have found that the Cartan subalgebra $\mathfrak h$ described is $2$-dimensional. To work out the root space decomposition, you must determine the eigenvalues and eigenvectors for the adjoint action of $\mathfrak h$ on $so(4)$, and to simplify matters you only need to see what happens for the $2$ basis elements of $\mathfrak h$. You should be able to work this out explicitly: take $E_1$ to be one basis element and write out $[E_1,X]$ for $X$ an arbitrary element of $so(4)$. (By the way, you should also be able to fully determine what an element of $so(4)$ looks like.) Then ask yourself when $[E_1,X]$ could possibly be a multiple of $X$ to find the eigenvalues and eigenvectors. You would do the same for the other basis element of $\mathfrak h$.

In the end you should find $4$ roots with a $1$-dimensional eigenspace for each, and so your root decomposition will look like $so(4) = \mathfrak h \oplus \mathfrak g_1 \oplus \mathfrak g_2 \oplus \mathfrak g_3 \oplus \mathfrak g_4$ where each $\mathfrak g_\alpha$ is $1$-dimensional. Finally you must figure out how to group these factors in the direct sum together to produce two copies of $sl(2)$ in order to produce the required isomorphism.

I can offer more details if you get stuck.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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