Weyl roots and equivalences of integral quadratic forms

Abstract

We study integral quadratic forms in the sense of Roiter, that is, quadratic forms whose integer coefficients satisfy certain divisibility condition assuring that the associated Weyl group is integral. Such forms are known to be useful for characterizing classes of finite-dimensional algebras and Lie algebras. We present a solution of the problem considered by Roiter in 1978 of providing a characterization of Weyl roots of positive definite forms which is independent on an explicit action of the Weyl group. We also prove some new facts on the Weyl group and the Weyl roots in general. Moreover, we provide a detailed comparison of several distinct equivalence relations of integral forms appearing in the literature. In particular, we discuss how these relations behave with respect to the Weyl roots and some properties of integral forms, and we supplement some missing proofs related to the classification of positive definite and principal forms.