Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-05-08003-2
Published electronically: July 19, 2005
MathSciNet review: 2180881
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. F. Cedó, E. Jespers and J. Okninski, Semiprime quadratic algebras of Gelfand-Kirillov dimension one, J. Algebra Appl. 3(2004), 283-300. MR 2096451
  • 2. P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100(1999), 169-209. MR 1722951 (2001c:16076)
  • 3. T. Gateva-Ivanova, E. Jespers and J. Okninski, Quadratic algebras of skew type and the underlying semigroups, J. Algebra 270(2003), 635-659. MR 2019633 (2004m:16039)
  • 4. T. Gateva-Ivanova and M. Van den Bergh, Semigroups of $I$-Type, J. Algebra 206(1998), 97-112. MR 1637256 (99h:20090)
  • 5. E. Jespers and J. Okninski, Quadratic algebras of skew type satisfying the cyclic condition, Int. J. Algebra and Computation 14(2004), 479-498. MR 2084381
  • 6. E. Jespers and J. Okninski, Monoids and groups of $I$-type, Algebras and Representation Theory, to appear.
  • 7. G.R. Krause and T.H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Revised edition. Graduate Studies in Mathematics, 22. American Mathematical Society, Providence, RI, 2000. MR 1721834 (2000j:16035)
  • 8. J. Tate and M. Van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124(1996), 619-647. MR 1369430 (98c:16057)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16P90, 16S36, 16S15, 20M25, 16P40, 20M05, 20F05

Retrieve articles in all journals with MSC (2000): 16P90, 16S36, 16S15, 20M25, 16P40, 20M05, 20F05


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

American Mathematical Society