Parameterizations of Infinite Biconvex Sets in Affine Root Systems

We investigate detailed relationships between the set W of all infinite “reduced” sequences of elements of a canonical generating set of an arbitrary affine Weyl group and the set B of all infinite sets having a certain “biconvex” property in a positive root system ∆+ of the corresponding untwisted affine Lie algebra g. The study is related to the classification of convex orders on ∆+, which are used to construct convex bases of Poincaré-Birkhoff-Witt type of the upper triangular subalgebra of the quantized enveloping algebra Uq(g). We construct a set P by using data of the underlying finite-dimensional simple Lie algebra, and bijective mappings ∇ : P → B and χ : P → W such that ∇ = Φ ◦ χ, where W is an quotient set of W and Φ : W → B is a natural injective mapping.