Integrating infinite-dimensional Lie algebras by a Tannaka reconstruction (Part I)

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 a certain category of duals. By a Tannaka reconstruction we associate to C and C a monoid M with a coordinate ring of matrix coefficients F [M ] (which has in general no natural coalgebra structure), as well as a Lie algebra Lie(M). The monoid M and the Lie algebra Lie(M) both act on the objects of C. We say that C and C du are good for integrating g, if the Lie algebra g is in a natural way a Lie subalgebra of Lie(M). 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 the coordinate ring F [M ] into the dual of the universal enveloping algebra U(g) of g, a Peter-and-Weyl-type theorem for F [M ] if C is a semisimple category, 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 and generalized toric submonoids of M . We show that C and C are good for integrating g if the Lie algebra g is generated by integrable locally finite elements. We interprete the monoid M algebraic geometrically as an irreducible weak algebraic monoid with Lie algebra Lie(M). The monoid M acts by morphisms of varieties on every object V of C. The action of the Lie algebra Lie(M) on V is the differentiated action. Mathematics Subject Classification 2000: 17B67, 22E65.