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
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
- Ferran Cedó, Eric Jespers, and Jan Okniński, Semiprime quadratic algebras of Gelfand-Kirillov dimension one, J. Algebra Appl. 3 (2004), no. 3, 283–300. MR 2096451, DOI 10.1142/S0219498804000848
- Pavel Etingof, Travis Schedler, and Alexandre Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), no. 2, 169–209. MR 1722951, DOI 10.1215/S0012-7094-99-10007-X
- T. Gateva-Ivanova, Eric Jespers, and Jan Okniński, Quadratic algebras of skew type and the underlying monoids, J. Algebra 270 (2003), no. 2, 635–659. MR 2019633, DOI 10.1016/j.jalgebra.2003.06.005
- Tatiana Gateva-Ivanova and Michel Van den Bergh, Semigroups of $I$-type, J. Algebra 206 (1998), no. 1, 97–112. MR 1637256, DOI 10.1006/jabr.1997.7399
- Eric Jespers and Jan Okniński, Quadratic algebras of skew type satisfying the cyclic condition, Internat. J. Algebra Comput. 14 (2004), no. 4, 479–498. MR 2084381, DOI 10.1142/S0218196704001864
- E. Jespers and J. Okniński, Monoids and groups of $I$-type, Algebras and Representation Theory, to appear .
- Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
- John Tate and Michel van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), no. 1-3, 619–647. MR 1369430, DOI 10.1007/s002220050065
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
- MR Author ID: 94560
- Email: efjesper@vub.ac.be
- Jan Okniński
- Affiliation: Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland
- Email: okninski@mimuw.edu.pl
- 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: https://doi.org/10.1090/S0002-9939-05-08003-2
- MathSciNet review: 2180881