Mesh geometries of root orbits of integral quadratic forms

Integral quadratic forms q:Zn→Z, with n≥1, and the sets Rq(d)={v∈Zn;q(v)=d}, with d∈Z, of their integral roots are studied by means of mesh translation quivers defined by Z-bilinear morsifications bA:Zn×Zn→Z of q, with Z-regular matrices A∈Mn(Z). Mesh geometries of roots of positive definite quadratic forms q:Zn→Z are studied in connection with root mesh quivers of forms associated to Dynkin diagrams An,Dn,E6,E7,E8 and the Auslander–Reiten quivers of the derived category Db(R) of path algebras R=KQ of Dynkin quivers Q. We introduce the concepts of a Z-morsification bA of a quadratic form q, a weighted ΦA-mesh of vectors in Zn, and a weighted ΦA-mesh orbit translation quiver Γ(Rq,ΦA) of vectors in Zn, where Rq≔Rq(1) and ΦA:Zn→Zn is the Coxeter isomorphism defined by A. The existence of mesh geometries on Rq is discussed. It is shown that, under some assumptions on the morsification bA:Zn×Zn→Z, the set admit a ΦA-orbit mesh quiver , where ΦA:Zn→Zn is the Coxeter isomorphism defined by A. Moreover, splits into three infinite connected components , , and , where are isomorphic to a translation quiver Z⋅Δ, with Δ an extended Dynkin quiver, and has the shape of a sand–glass tube.