仿射根系中无限双凸集的参数化
Parametrizations of infinite biconvex sets in affine root systems
摘要：
本文详细研究了任意无扭仿射李代数${\mathfrak g}$的正根系$Δ_+$中所有无限“双凸”集的集合${\mathfrak B}^\infty$与${\mathfrak g}$的Weyl群的所有无限"约化字“集${\mathcal W}^\infty$之间的关系.该研究应用于$Δ_+$上的“凸序”分类（参见。\cite{kI}），它们是构造量子化泛包络代数$U_q({\mathfrak g})$的上三角子代数$U_q^+$的Poincaré-Birkhoff-Witt型“凸基”所必需的.我们利用有限维单李代数的数据和双射映射$\nabla\colon\boldsymbol{\mathcal P}\to{\mathfrak B}^\infty$和$χ\colon\boldsymbol{\mathcal P}\to W^\infty$构造了一个集合$\boldsymbol{\mathcal P}$，使得$\nabla=\varPhi^\infty\circχ$，其中$W^\infty$是${\mathcal W}^\infty$的商集，$\varPhi^\infty\colon W^\infty\to{\mathfrak B}^\infty$是自然单射映射.
英文摘要：
We investigate in detail relationships between the set ${\mathfrak B}^\infty$ of all infinite ``biconvex'' sets in the positive root system $Δ_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and the set ${\mathcal W}^\infty$ of all infinite ``reduced word'' of the Weyl group of ${\mathfrak g}$. The study is applied to the classification of ``convex orders'' on $Δ_+$ (cf. \cite{kI}), which are indispensable to construct ``convex bases'' of Poincaré-Birkhoff-Witt type of the upper triangular subalgebra $U_q^+$ of the quantized universal enveloping algebra $U_q({\mathfrak g})$. We construct a set $\boldsymbol{\mathcal P}$ by using data of the underlying finite-dimensional simple Lie algebra, and bijective mappings $\nabla\colon\boldsymbol{\mathcal P}\to{\mathfrak B}^\infty$ and $χ\colon\boldsymbol{\mathcal P}\to W^\infty$ such that $\nabla=\varPhi^\infty\circχ$, where $W^\infty$ is an quotient set of ${\mathcal W}^\infty$ and $\varPhi^\infty\colon W^\infty\to{\mathfrak B}^\infty$ is a natural injective mapping.
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