The Gelfand-kirillov Dimension of Quadratic Algebras Satisfying the Cyclic Condition

We consider algebras over a field K presented by generators x1, . . . , xn and subject to (n 2 ) 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 x1, . . . , 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.