• 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.

## References

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)
3. 8.
Barot, M., de la Peña, J.A.: Algebras whose Euler form is non-negative. Colloq. Math. 79, 119–131 (1999)
4. 9.
Barot, M., de la Peña, J.A.: Root-induced integral quadratic forms. Linear Algebra Appl. 412, 291–302 (2006)
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)
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)
9. 52.
von Höhne, H.-J.: On weakly positive unit forms. Comment. Math. Helvetici 63, 312–336 (1988)

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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