Root strings with three or four real roots in Kac-Moody root systems
Jun Morita · Tohoku Mathematical Journal · 1988
A characterization and a presentation of a (universal) Kac-Moody group over a field (of any characteristic) have been given by Tits [6].Such a presentation, which is a natural generalization of Steinberg's one for a (simply connected) split semisimple algebraic group over a field (cf.[5]), is conjectured by E. Abe and established by J. Tits.The most interesting part of the presentation is the so-called "commutation relation", which is deeply related to the root strings and whose explicit description is given in [4].In this paper, we will discuss certain root strings in Kac-Moody root systems, and give some direct applications to the associated Kac-Moody groups.Our main result is as follows.Let A = (α,y) be an nxn generalized Cartan matrix, A the associated root system, and A τe the set of real roots.Put r(α; β) = # \{β + ka \ k e Z}f] A τe \ for (α, β)eA τe xA.Then the following two conditions are equivalent.(1) r(a; β) = 3 or 4 for some (α, β) e A τe x A.(2) a iS = -1 and a H < -1 for some i, j (1 ^ i, j ^ n).As a corollary, we can simplify the Steinberg-Tits presentation of the associated Kac-Moody group in the case when A has a certain property.