Universal Enveloping Algebra U(g) is noetherian

Universal Enveloping Algebra $U(\mathfrak{g})$ is noetherian
證明大綱:

這有什麼好處?



這樣的話 category $\mathcal{O}$ 才可以被證明是 noetherian
(即任何 module $M$ $\in$ $\mathcal{O}$ 都是 noetherian module)
since $\mathcal{O}$ 中 module $M$ is finitely generated as a $U(\mathfrak{g})$-module

又可以證明 category $\mathcal{O}$ is artinian

既 artinian 又 noetherian
就可以套用 Krull-Remak-Schmidt 定理
將 $M$ 拆成 finite direct sum of indecomposable submodules
          uniquely determined up to permutations !


這樣我們就可以將 category $\mathcal{O}$ 拆成 subcategory $\mathcal{O}_\chi$ related to central character $\chi$!