Quadratic algebras of skew type and the underlying monoids

Abstract

We consider algebras over a field K defined by a presentation K〈x1,…,xn∣R〉, where R consists of n2 square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand–Kirillov dimension defined by homogeneous semigroup relations satisfy a polynomial identity. The structure of the underlying monoid, defined by the same presentation, is described. This is used to derive information on the prime radical and minimal prime ideals. Some examples are described in detail. Earlier, Gateva-Ivanova and van den Bergh, and Jespers and Okniński considered special classes of such algebras in the contexts of noetherian algebras, Gröbner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang–Baxter equation.