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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: July 19, 2005
MathSciNet review: 2180881
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Abstract: We consider algebras over a field $K$ presented by generators $x_1,\dots ,$ $x_n$ and subject to $n\choose 2$ square-free relations of the form $x_{i}x_{j}=x_{k}x_{l}$ with every monomial $x_{i}x_{j}, i\neq j$, 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 $n\geq 4$, we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators $x_1,\dots ,x_n$ 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 $x_1,\dots ,x_n$ 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: http://dx.doi.org/10.1090/S0002-9939-05-08003-2
PII: S 0002-9939(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