On the corank of the Tits form of a tame algebra

Abstract

Let Λ = k[Q] I be a finite-dimensional, directed k-algebra with k an algebraically closed field. Let q Λ be the Tits (quadratic) form of Λ. The isotropic corank of q Λ denoted by corank 0 q gL , is the maximal dimension of a convex half-space over Q contained in Σ 0( q Λ = {0 ≤ ν ϵ Q n : q Λ(ν) = 0}, where n is the number of vertices of Q. We show that a strongly simply connected cycle-finite algebra Λ, has corank 0 qΛ ≤ 2. A strongly simply connected algebra Λ is tame domestic if and only if q gL is weakly non-negative and corank 0 q Λ ≤ 1. We also characterize polynomial growth algebras using invariants associated with the Tits form.