Root-induced integral quadratic forms

Abstract

Given an integral quadratic unit form q : Z n → Z and a finite tuple of q-roots r = ( r j ) j∈ J the induced q-root form q r is considered as in [P. Gabriel, A.V. Roiter, Representations of finite dimensional algebras, in: A.I. Kostrikin, I.V. Shafarevich (Eds.), Algebra VIII, Encyclopaedia of the Mathematical Sciences, vol. 73, 1992, Springer (Chapter 6)]. We show that two non-negative unit forms are of the same Dynkin type precisely when they are root-induced one from the other. Moreover, there are only finitely many non-negative unit forms without double edges of a given Dynkin type. Root-induction yields an interesting partial order on the Dynkin types, which is studied in the paper.