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

Authors:
Ferran Cedó, Eric Jespers and Jan Okninski

Journal:
Proc. Amer. Math. Soc. **134** (2006), 653-663

MSC (2000):
Primary 16P90, 16S36, 16S15, 20M25; Secondary 16P40, 20M05, 20F05

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

Published electronically:
July 19, 2005

MathSciNet review:
2180881

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Abstract: We consider algebras over a field presented by generators and subject to square-free relations of the form with every monomial , appearing in one of the relations. It is shown that for 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 . For , we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators has Gelfand-Kirillov dimension if and only if it is of -type, and this occurs if and only if the multiplicative submonoid generated by is cancellative.

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Additional Information

**Ferran Cedó**

Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Email:
cedo@mat.uab.es

**Eric Jespers**

Affiliation:
Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium

Email:
efjesper@vub.ac.be

**Jan Okninski**

Affiliation:
Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland

Email:
okninski@mimuw.edu.pl

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

Received by editor(s):
March 24, 2004

Received by editor(s) in revised form:
October 19, 2004

Published electronically:
July 19, 2005

Additional Notes:
This work was supported in part by the Flemish-Polish bilateral agreement BIL 01/31 and KBN research grant 2P03A 033 25 (Poland), the MCyT-Spain and FEDER through grant BFM2002-01390, and by the Generalitat de Catalunya (Grup de Recerca consolidat 2001SGR00171).

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2005
American Mathematical Society