The root system of sign \((1,0,1)\) (Q1065138)

Let \(F\) be a real vector space equipped with a quadratic form \(q\), whose signature is \((\mu_+,\mu_ 0,\mu_-)\) \((\mu_+\) and \(\mu_-\) are maximal ranks of linear subspaces of \(F\) on which \(q\) is positive or negative definite respectively. \(\mu_ 0\) is the rank of the radical of \(q\).) A subset \(R\) of \(F\) is called a root system of sign \((\mu_+,\mu_ 0,\mu_-)\) if it satisfies usual axioms of a root system similar to the classical one. In this note, the author gives a classification of root systems of sign \((1,0,1)\). There are 72 types of root systems. Each of them contains an infinite sequence of root systems, whose isomorphism classes are parametrized by a positive integer called the period of \(R\), and finite numerical data called the coefficients of the diagram of the type. For a root system \(R\), the Weyl group, i.e. the group generated by reflections of elements of \(R\) is calculated, whose invariants are elementary hyperbolic functions. As an example, the author shows that the unit group of a real quadratic field is a root system of sign \((1,0,1)\). Also, as an application of the study of root systems, the author classifies indefinite quadratic forms defined on a \(Z\)-free module of rank 2 from a viewpoint of a maximal root system belonging to the quadratic form.