Lie group (additional condition)

[namlessom]

Does somebody know an example of a differentiable manifold which is a group but NOT a Lie group? So the additional condition: the group operations multiplication and inversion are analytic maps, is not satisfied.

 

[matt grime]

Yes, but they aren't particularly natural.

Let M be any differential manifold. Let G be any group such that card(M)=card(G). Let f be any bijection of the underlying sets. Then F makes M into a group by fiat, and in general, if we pick f in completely arbitrary fashion, it will not be a lie group.

More natural ones do not immediately spring to mind, sorry.

 

[M98Ranger]

One example of a differentiable manifold that is a group but not a Lie group is the Heisenberg group. The Heisenberg group is a three-dimensional manifold that is the set of 3x3 upper triangular matrices with real entries, where the group operation is matrix multiplication. However, the group operations of multiplication and inversion are not analytic maps, as they involve taking square roots of negative numbers, which is not allowed in real analytic functions. Therefore, the Heisenberg group satisfies the conditions of a differentiable manifold and a group, but not the additional condition of having analytic group operations, making it a non-Lie group.