Realizing enveloping algebras via varieties of modules

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