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Nonnegative Quadratic Forms

  • Michael Barot
  • Jesús Arturo Jiménez González
  • José-Antonio de la Peña
Chapter
Part of the Algebra and Applications book series (AA, volume 25)

Abstract

Let G be a connected graph. In this Chapter we show that qG is a nonpositive, nonnegative unit form if and only if G is an extended Dynkin diagram, and give a short proof for Vinberg’s characterization of such diagrams. As shown by Barot and de la Peña, for a nonnegative semi-unit form q there is an iterated flation T such that qT = qΔ ⊕ ξc, where Δ is a disjoint union of Dynkin diagrams and ξc is the zero quadratic form in c variables. The uniquely determined union of Dynkin diagrams Δ is referred to as the Dynkin type of q, while c is the corank of q (the rank of the radical of q). Hypercritical nonnegative unit forms are considered (those borderline forms between positive and nonnegative forms), and a characterization of such forms is provided. We say that two unit forms q and q′ are root equivalent if q is a q′-root induced form, and q′ is a q-root induced form. Here we show that two non-negative semi-unit forms have the same Dynkin type if and only if they are root equivalent, and derive an interesting partial order in the set of Dynkin types.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Michael Barot
    • 1
  • Jesús Arturo Jiménez González
    • 2
  • José-Antonio de la Peña
    • 3
  1. 1.Kantonsschule SchaffhausenSchaffhausenSwitzerland
  2. 2.Instituto de MatemáticasUNAMMexico CityMexico
  3. 3.Instituto de MatemáticasMiembro de El Colegio NacionalUNAMMexico

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