Nonnegative Quadratic Forms

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


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.


  1. 3.
    Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1997)Google Scholar
  2. 7.
    Barot, M., de la Peña, J.A.: The Dynkin type of a non-negative unit form. Expo. Math. 17, 339–348 (1999)MathSciNetzbMATHGoogle Scholar
  3. 8.
    Barot, M., de la Peña, J.A.: Algebras whose Euler form is non-negative. Colloq. Math. 79, 119–131 (1999)MathSciNetCrossRefGoogle Scholar
  4. 9.
    Barot, M., de la Peña, J.A.: Root-induced integral quadratic forms. Linear Algebra Appl. 412, 291–302 (2006)MathSciNetCrossRefGoogle Scholar
  5. 32.
    Happel, H., Preisel, U., Ringel, C.M.: Vinberg’s characterization of Dynkin diagrams using subadditive functions with applications to DTr-periodic modules. In: Proceedings of the ICRA II, Ottawa 1979. Lecture Notes in Math, vol. 832, pp. 280–294. Springer, Berlin, Heidelberg, New York (1980)CrossRefGoogle Scholar
  6. 43.
    Ovsienko, A.: Integral weakly positive forms (Russian). In: Schurian Matrix Problems and Quadratic Forms, pp. 3–17. Mathematics Institute of the Academy of Sciences of the Ukrainian SSR, Kiev (1978)Google Scholar
  7. 46.
    Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, New York (1984)Google Scholar
  8. 51.
    Vinberg, E.B.: Discrete linear groups generated by reflections. Izv. Akad. Nauk SSSR 35 (1971). Transl.: Math. USSR Izvestija 5, 1083–1119 (1971)CrossRefGoogle Scholar
  9. 52.
    von Höhne, H.-J.: On weakly positive unit forms. Comment. Math. Helvetici 63, 312–336 (1988)MathSciNetCrossRefGoogle Scholar

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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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