Realizing Enveloping Algebras via Varieties of Modules

Abstract

By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra $L(\hat{I}›)$ generated by indecomposable constructible sets in the varieties of modules for any finite dimensional $\mathbb{C}$-algebra $\hat{I}›.$ We obtain a geometric realization of the universal enveloping algebra $R(\hat{I}›)$ of $L(\hat{I}›).$ 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 $\hat{I}›$ and use it to give the comultiplication formula in $R(\hat{I}›).$