QUADRATIC ALGEBRAS OF SKEW TYPE SATISFYING THE CYCLIC CONDITION

Abstract

We consider algebras over a field with generators x1,x2,…,xn subject to [Formula: see text] square-free relations xixj=xkxl in which every product xpxq, p≠q, appears in one of the relations. The work of Gateva-Ivanova and Van den Bergh, motivated in particular by the study of set theoretic solutions of the Yang–Baxter equation, provided an important class of such algebras. In this special case the presentation satisfies the so-called cyclic condition that became an essential combinatorial tool in proving that these algebras share many strong ring theoretic properties of polynomial algebras in commuting variables. In this paper we describe the structure of algebras on four generators satisfying the cyclic condition. The emphasis is on some new unexpected features, not present in the motivating special classes.