TY - JOUR

T1 - The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition

AU - Cedó, Ferran

AU - Jespers, Eric

AU - Okniński, Jan

PY - 2006/3/1

Y1 - 2006/3/1

N2 - 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.

AB - 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.

U2 - https://doi.org/10.1090/S0002-9939-05-08003-2

DO - https://doi.org/10.1090/S0002-9939-05-08003-2

M3 - Article

VL - 134

SP - 653

EP - 663

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

ER -