6
$\begingroup$

While studying the book Introduction to Lie Algebras and Representation Theory, by Humphreys, I've came across a problem that seems simple, but I just cannot figure out:

If $\alpha, \beta \in \Phi$, where $\Phi$ is the root system, then $(\alpha + \beta)^\vee = \alpha ^\vee + \beta^\vee$?

Studying the algebras I already know it feels true, but I wasn't able to show a proper proof.

Any help would be appreciated.

$\endgroup$
7
$\begingroup$

If this were true, a root system would be isomorphic to its dual, which need not be the case. Look for a counterexample in type $B$.

$\endgroup$

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.