Is there a finite dimensional Lie algebra L such that there are infinite number of non isomorphic compact connected lie groups which Lie algebras are isomorphic to L?
Asked
Active
Viewed 200 times
1
-
5$\mathfrak{sl}_2(\mathbf{R})$ – YCor Jan 20 '14 at 10:09
-
6I do not think it is a good idea to edit the question (without any mention of the edit), especially after the answer appeared. – Sasha Anan'in Jan 20 '14 at 11:07
1 Answers
10
To develop Yves' comment : let $G$ be the simply connected Lie group with Lie algebra $\mathfrak{sl}_2(\mathbb{R})$; it contains a central subgroup $Z\cong \mathbb{Z}$ such that $G/Z\cong \mathrm{SL}_2(\mathbb{R})$. Now put $G_n:=G/nZ$ for $n\geq 1$. An isomorphism $G_p\rightarrow G_q$ lifts to an isomorphism $G\rightarrow G$ which must map $pZ$ into $qZ$; this implies $p=q$, thus all these groups are non-isomorphic.
-
thank you very much for the answer. What about if we require that all groups are compact, as I revised the question in this new version? – Jan 20 '14 at 11:02
-
4Then the answer is negative. A compact Lie group admits a finite covering $T\times S$, with $T$ a torus and $S$ semi-simple, and these have only a finite number of non-isomorphic quotients. – Jan 20 '14 at 11:19