Realizing Enveloping Algebras via Varieties of Modules

Details Realizing Enveloping Algebras via Varieties of Modules Realizing Enveloping Algebras via Varieties of Modules

2006-04-26

arxiv.org/abs/math/0604560v3

Mathematics

Quantum Algebra

Scientific paper

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 $R(\Lambda).$

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