Infinite number of Lie groups with the same lie algebra

Question

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?

Answer 1

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.