Parametrizations of Infinite Biconvex Sets in Affine Root Systems

We investigate in detail relationships between the set B of all infinite “biconvex” sets in the positive root system ∆+ of an arbitrary untwisted affine Lie algebra g and the set W of all infinite “reduced word” of the Weyl group of g. The study is applied to the classification of “convex orders” on ∆+ (cf. [Ito]), which are indispensable to construct “convex bases” of PoincaréBirkhoff-Witt type of the upper triangular subalgebra U q of the quantized universal 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. Mathematics Subject Classifications (2000). 20F55, 17B37.