Realizing enveloping algebras via varieties of modules

By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra L(.) generated by indecomposable constructible sets in the varieties of modules for any finite-dimensional .-algebra .. We obtain a geometric realization of the universal enveloping algebra R(.) of L(.), this generalizes the main result of Riedtmann. We also obtain Green.s formula in a geometric form for any finite-dimensional .-algebra . and use it to give the comultiplication formula in R(.).