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Mesh Geometries of Root Orbits of Integral Quadratic Forms by Daniel Simson is a Mathematics article available to read on EtoBox.

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Integral quadratic forms q : Z n → Z, with n ≥ 1, and the sets R q (d) = {v ∈ Z n ; q(v) = d}, with d ∈ Z, of their integral roots are studied by means of mesh translation quivers defined by Z-bilinear morsifications b A : Z n × Z n → Z of q, with Z-regular matrices A ∈ M n (Z). Mesh geometries of roots of positive definite quadratic forms q : Z n → Z are studied in connection with root mesh quivers of forms associated to Dynkin diagrams A n , D n , E 6 , E 7 , E 8 and the Auslander-Reiten quivers of the derived category D b (R) of path algebras R = KQ of Dynkin quivers Q . We introduce the concepts of a Z-morsification b A of a quadratic form q, a weighted Φ A -mesh of vectors in Z n , and a weighted Φ A -mesh orbit translation quiver Γ (R q , Φ A ) of vectors in Z n , where R q := R q (1) and Φ A : Z n → Z n is the Coxeter isomorphism defined by A. The existence of mesh geometries on R q is discussed. It is shown that, under some assumptions on the morsification b A : is the Coxeter isomorphism defined by A. Moreover, Γ (R q ∪ Ker q, Φ A ) splits into three infinite connected components Γ (∂ - A R q ), Γ (∂ + A R q ), and Γ (∂ 0 A R q ∪ Ker q), where Γ (∂ - A R q ) ∼ = Γ (∂ + A R

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Author

Daniel Simson

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0022-4049)

Published

2011

Language

EN

Field

Mathematics (Physical Sciences)