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 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 $\mathcal{P}$ by using data of the underlying finite-dimensional simple Lie algebra, and bijective mappings $\nabla :\mathcal{P}\to {\mathfrak{B}}^{\infty }$ and $\chi :\mathcal{P}\to W^{\infty }$ such that $\nabla ={\Phi }^{\infty }\circ \chi$, where $W^{\infty }$ is an quotient set of ${\mathcal{W}}^{\infty }$ and ${\Phi }^{\infty }:W^{\infty }\to {\mathfrak{B}}^{\infty }$ is a natural injective mapping.

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