Schurian vector space categories, hereditary algebras and Roiter's norm

Abstract

We associate a positive real number {norm of matrix} C {norm of matrix} to any vector space K-category C over a field K. Generalizing a result of Nazarova and Roiter, we show that a schurian vector space K-category C is representation-finite if and only if ind C is finite and {norm of matrix} C {norm of matrix} > frac(1, 4). Such vector space categories are quasilinear, i.e. its indecomposables are simple modules over their endomorphism ring. Recently, Nazarova and Roiter introduced the concept of P-faithful poset in order to clarify the structure of critical posets. Their conjecture on the precise form of P-faithful posets was established by Zeldich. We generalize these results and characterize P-faithful quasilinear vector space K-categories in terms of a class of hereditary algebras Hρ (D) parametrized by a skew-field D and a rational number ρ ≥ 1. © 2006 Elsevier Inc. All rights reserved.