Convex orderings in affine root systems
Let \(\Delta^0\) be a root system of a finite-dimensional complex simple Lie algebra \({\mathfrak g}^0\). Fix a positive system \(\Delta^0_+\) in \(\Delta^0\). The author [Proc. Am. Math. Soc. 120, 661-665 (1994; Zbl 0799.20037)] gave the definition of a compatible subset \(L\) of \(\Delta^0_+\) and proved that a subset \(L\) of \(\Delta^0_+\) is compatible if and only if \(L\) is associated to a unique element \(w\) in the Weyl group of \(\Delta^0\), moreover, the ordering on a totally ordered subset \(L \subseteq \Delta^0_+\) depends on the reduced expression of \(w\). In this paper, the definition of a compatible subset \(L\) of \(\Delta_+^{re}\) has been given where \(\Delta_+^{re}\) is the set of positive real roots of an affine Kac-Moody Lie algebra \({\mathfrak g}\), and it is proved that the results for \({\mathfrak g}\) are analogous to those for \({\mathfrak g}^0\).
scientific article; zbMATH DE number 763920