Weak Dynkin type and the universality of non-negative Coxeter-regular integral quadratic forms

An integral quadratic form is called Coxeter-regular if its integer coefficients satisfy a divisibility condition equivalent to the fact that the associated Coxeter transformation and Weyl group are integral. Such forms are known to be useful in the study of finite-dimensional associative algebras, Lie algebras and certain singularities. We show that a non-negative (connected) Coxeter-regular form q is universal (that is, q represents all non-negative integers) if and only if q represents the integers 1,2,3,7 and 14 . This may be viewed as a specialization (and, actually, an extension) of the Conway–Schneeberger/Bhargava “15 Theorem”. As one of the main tools we provide a complete classification, up to \mathbb{Z} -equivalence, of all non-negative Coxeter-regular forms by means of so-called weak Dynkin type, which is a certain equivalence class of a Dynkin (bi)graph. In this way, we obtain a generalization of the known result of Barot-de la Peña for unit forms and simply-laced Dynkin diagrams.