Let u : G → A be a
differentiable representation of a Lie group into a b-algebra. The differential u0= due
of u at the neutral element e of G is a representation of the Lie algebra g of G into A.
Because a Lie group is locally the union of one-parameter subgroups and since the
infinitesimal generator of a differentiable (multiplicative) sub-semi-group of A
determines this sub-semi-group, the representation u0 determines u if G is
connected.
We shall be concerned with the converse: given a representation u0 of g, when can
it be obtained by differentiating a representation u of G? We shall assume G
connected and simply connected, which means that we are only interested in the local
aspect of the problem.
