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

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?

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See the related MathOverflow question: mathoverflow.net/questions/44060/…. The answer is yes. – Jonas Meyer Jan 11 '11 at 4:47
Ah, thanks, Jonas. – Jesse Madnick Jan 11 '11 at 5:07