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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition
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by Ferran Cedó, Eric Jespers and Jan Okniński PDF
Proc. Amer. Math. Soc. 134 (2006), 653-663 Request permission


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:
  • Eric Jespers
  • Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
  • MR Author ID: 94560
  • Email:
  • Jan Okniński
  • Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
  • Email:
  • 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
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 653-663
  • MSC (2000): Primary 16P90, 16S36, 16S15, 20M25; Secondary 16P40, 20M05, 20F05
  • DOI:
  • MathSciNet review: 2180881