Parametrizations of infinite biconvex sets in affine root systems

Abstract

We investigate in detail relationships between the set ${\mathfrak B}^\infty$ of all infinite ``biconvex'' sets in the positive root system $\Delta_+$ 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 $\Delta_+$ (cf. \cite{kI}), which are indispensable to construct ``convex bases'' of Poincar\'e-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 $\chi\colon\boldsymbol{\mathcal P}\to W^\infty$ such that $\nabla=\varPhi^\infty\circ\chi$, 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.