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?


migrated from mathoverflow.net Jan 20 '14 at 14:48

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    $\begingroup$ $\mathfrak{sl}_2(\mathbf{R})$ $\endgroup$ – YCor Jan 20 '14 at 10:09
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    $\begingroup$ I do not think it is a good idea to edit the question (without any mention of the edit), especially after the answer appeared. $\endgroup$ – Sasha Anan'in Jan 20 '14 at 11:07

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.

  • $\begingroup$ 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? $\endgroup$ – Ali Taghavi Jan 20 '14 at 11:02
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    $\begingroup$ Then 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. $\endgroup$ – abx Jan 20 '14 at 11:19

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