Realizing Enveloping Algebras via Varieties of Modules

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