Quadratic algebras of skew type and the underlying semigroups

Abstract

We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from 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 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, Etingof, Schedler and Soloviev, Gateva-Ivanova and Van den Bergh, and the authors considered special classes of such algebras in the contexts of noetherian algebras, Grobner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang-Baxter equation.