A notion of rank for noncommutative quadratic forms on four generators

Abstract

In this paper, we extend work from [M. Vancliff and P. P. Veerapen, Generalizing the notion of rank to noncommutative quadratic forms, in Noncommutative Birational Geometry, Representations and Combinatorics, eds. A. Berenstein and V. Retakh, Contemporary Mathematics, Vol. 592 (2013), pp. 241–250], where a notion of rank, called [Formula: see text]-rank, was proposed for noncommutative quadratic forms on two and three generators. In particular, we provide a definition of [Formula: see text]-rank one and two for noncommutative quadratic forms on four generators. We apply this definition to determine the number of point modules over certain quadratic AS-regular algebras of global dimension four.