Affine root systems and dual numbers

Extended Affine Root Systems

There are two notions of the extended affine root systems in the literature which both are introduced axiomatically. One, extended affine root system (SAERS for short), consists only of nonisotropic roots, while the other, extended affine root system (EARS for short), contains certain isotropic roots too. We show that there is a one to one correspondence between (reduced) SEARSs and EARSs. Name...

Root numbers

Finite Fields, Root Systems and Orbit Numbers of Chevalley Groups

We describe combinatorial techniques to determine the numbers of semisim-ple conjugacy classes and adjoint orbits with xed class of centralizers for simply connected nite groups of Lie type.

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 typ...

The Classification of Convex Orders on Affine Root Systems

We classify all total orders having a certain convexity property on the positive root system of an arbitrary untwisted affine Lie algebra g. Such total orders are called convex orders and were used to construct Poincaré-BirkhoffWitt type bases of the upper triangular subalgebra of the quantized enveloping algebra of g which are called convex bases.