0

Can the centre of a (compact, semisimple) Lie group $G$ be discrete? If yes, under which conditions on $G$ this happens? Any references would be appreciated.

user3257624
  • 361

1 Answers1

3

If $G$ is a compact semisimple Lie group, then its center $Z_G$ is always discrete. This is because if $\mathfrak{g}$ is the Lie algebra of $G$, then $$\mathrm{Lie}(Z_G)=Z_{\mathfrak{g}},$$ the center of $\mathfrak{g}$. But $Z_{\mathfrak{g}}=\{0\}$ since $\mathfrak{g}$ is semisimple. (This is because $Z_{\mathfrak{g}}$ is an abelian ideal, so also a solvable ideal.)

Spenser
  • 19,469
  • Thank's! Do you have any reference in mind? – user3257624 Mar 12 '17 at 10:55
  • 1
    @user3257624 I don't know if you can find that specific statement, but a general good reference for Lie groups and Lie algebras is Knapp Lie groups beyond an introduction. – Spenser Mar 12 '17 at 10:59
  • Would that be Proposition 7.9? – user3257624 Mar 12 '17 at 11:11
  • @user3257624 That's a much stronger result with strong assumptions. Why do you need a reference? It's better to just understand the simple argument above there. Every ingredient used in that proof can be found in the book though. Is there some part you don't understand? – Spenser Mar 12 '17 at 11:25
  • 1
    @user3257624 You can find the statement as a remark below Proposition 6.30 on page 361. – Spenser Mar 12 '17 at 11:28
  • 1
    @user3257624 Note that we don't need $G$ to be compact. But if it is compact, then $Z_G$ is actually finite. This is because a closed subgroup of a compact space is compact and a compact discrete space is finite. – Spenser Mar 12 '17 at 11:31
  • Great, I found that remark! Thank you very much. I need a refernce for my thesis and I don't want to prove the statement myself, because my topic is not reated to Group theory. – user3257624 Mar 12 '17 at 11:34