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.

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

Suppose some maximal torus $T$ of $G$ is $C_G(T)$, then the set of semisimple elements for which $C_G(s)$ is a torus contains a nonempty open set. Such element are called regular semisimple.

I want to know how to prove this claim and want to find some good examples for some common algebraic groups such as $\mathrm{SL}(n,K)$.

share|cite|improve this question
If $K$ in the question is algebraically closed then any semisimple element $s$ can be diagonalized; it is regular if and only if all eigenvalues are distinct. This is equivalent, as Turgeon refers to Borel's book, to being outside the kernel of every root. – Abhishek Parab Jun 27 '15 at 15:50
up vote 0 down vote accepted

(I will assume you are still interested in getting an answer to this problem.)

By Lemma 12.2 of Borel's Linear Algebraic Groups, being regular is equivalent to being fixed by no roots. Therefore, the set of regular elements is a finite intersection of open sets, and so it is open.

share|cite|improve this answer

Your Answer


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.