  Let g be a Lie algebra over a field F of characteristic zero, let C be a
certain tensor category of representations of g, and C-du a certain category of
duals. By a Tannaka reconstruction we associate to C and C-du a monoid M with
coordinate ring of matrix coefficients F[M], (which has in general no natural
coalgebra structure), as well as a Lie algebra Lie(M). We interprete the monoid
M algebraic geometrically as a weak algebraic monoid with Lie algebra Lie(M).
The monoid M acts by morphisms of varieties on every object of C, the Lie
algebra Lie(M) acts by the differentiated action.
  If g is in a natural way a subalgebra of Lie(M), we say that C and C-du are
good for integrating g. In this situation we treat: The adjoint action of the
unit group of M on Lie(M). The relation between g and M-invariant subspaces.
The embedding of F[M] in the dual of the universal enveloping algebra of g. A
Peter-and-Weyl-type theorem if the category C is semisimple. The
Jordan-Chevalley decompositions for certain elements of M and Lie(M). An
embedding theorem related to subalgebras of g which act locally finite.
Prounipotent subgroups of M, and generalized toric submonoids of M. We show
that C and C-du are good for integrating g, if the Lie algebra g is generated
by integrable locally finite elements.
