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Realizing Enveloping Algebras via Varieties of Modules

Ming Ding·Jie Xiao·Fan Xu
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Abstract

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 $\mathbb{C}$-algebra $Λ.$ We obtain a geometric realization of the universal enveloping algebra $R(Λ)$ of $L(Λ).$ This generalizes the main result of Riedtmann in \cite{R}. We also obtain Green's theorem in \cite{G} in a geometric form for any finite dimensional $\mathbb{C}$-algebra $Λ$ and use it to give the comultiplication formula in $R(Λ).$

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Ming Ding, Jie Xiao, Fan Xu. 2009-02-03. Realizing Enveloping Algebras via Varieties of Modules. https://arxiv.org/abs/math/0604560

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