Let $\mathfrak{g}$ be a Lie algebra with the Cartan matrix $$ C=\left(\begin{array}{cc} 2 & -2\\ -1 & 2 \end{array}\right) $$


  1. How can the number of roots of $\mathfrak{g}$ be determined from $C$?

  2. How can it be shown that the dimension of $\mathfrak{g}$ is 10?

Note: this is an exam prep question.


The number of roots can be found by using the Cartan matrix to find the type of the root system. In this case the type is $B_2$ (or $C_2$), so there are $8$ roots.

The Lie algebra is the direct sum of a Cartan subalgebra and the weight spaces. In this case the Cartan subalgebra is two dimensional and the 8 weight spaces are one dimensional, so the dimension is 10.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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