We consider algebras over a field K presented by generators x 1,..., xn and subject to (n2) square-free relations of the form xixj = xkxl with every monomial xixj, i ≠ j, appearing in one of the relations. It is shown that for n > 1 the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condition. It is known that this dimension is an integer not exceeding n. For n ≥ 4, we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators x 1,...,xn has Gelfand-Kirillov dimension n if and only if it is of I-type, and this occurs if and only if the multiplicative submonoid generated by x1,...,xn is cancellative. © 2005 American Mathematical Society.
|Journal||Proceedings of the American Mathematical Society|
|Publication status||Published - 1 Mar 2006|