Diffeomorphic, group-isomorphic Lie groups that are not isomorphic as Lie groups

Solution 1:

Just to get this one off of the unanswered list, I'll post Jonas's link as a Community Wiki answer:

The answer is yes, there do exist such Lie groups: here is a MathOverflow question with the details.

Solution 2:

The examples (from the mathoverflow link) in the accepted answer are not quite Lie groups (at least not with the standard definition): They are not 2nd countable. However, there are examples of simply connected nilipotent Lie groups which are isomorphic as abstract groups but not as Lie groups, see here. These groups are both diffeomorphic to ${\mathbb C}^7$. At the same time, if you restrict to the class of semisimple Lie groups then an abstract isomorphism implies the existence of an isomorphism as Lie groups. (Although, the given abstract isomorphism may fail to be continuous.)