Do there exist two Lie groups which are diffeomorphic as smooth manifolds, have isomorphic group structures, yet are not isomorphic as Lie groups?

Of course, for this to happen, any diffeomorphism would fail to preserve the group structure, and any group isomorphism would either fail to be smooth, or its inverse would fail to be smooth.

I have no other reason for asking this other than out of curiosity. (In particular, this is not a problem I found out of a textbook.)

Related Question: Are there topological groups that are homeomorphic and have isomorphic group structures, yet are not isomorphic as topological groups?


2 Answers 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.)


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.


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