Hilbert’s 5th Problem

Algebra × Topology = Analysis Important Lie groups are the vector groups Rn, their compact quotients Rn/Zn, the general linear groups GLn(R), and the orthogonal groups On(R). For each of these the group structure and the real analytic manifold structure is the obvious one; for example, GLn(R) is open as a subset of Rn 2 , and thus an open submanifold of the analytic manifold Rn2 . Hilbert’s 5th problem asks for a characterization of Lie groups that is free of smoothness or analyticity requirements. A topological group is said to be locally euclidean if some neighborhood of its identity is homeomorphic to some Rn. A Lie group is obviously locally euclidean, and the most common version of Hilbert’s 5th problem (H5) can be stated as follows: