Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

Question

I've learned that a compact connected abelian Lie group must be a torus. Of course, conversely, a torus as a group is abelian.

I wonder if 'homeomorphic to a torus' is enough to imply abelian.

Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

Answer 1

In the compact case, the cohomology of a lie group coincides with the cohomology of it's lie algebra in particular $H^1(\frak g ) = \frak g / [ \frak g , \frak g ] $, since this has rank n for an n torus we have $\dim [ \frak g , \frak g ] = 0 $, so $G$ must be abelian.