Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-18T02:14:17.730Z Has data issue: false hasContentIssue false
  • English
  • Français

Gelfand–Kirillov dimension of differential difference algebras

Part of: Rings and algebras arising under various constructions Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  01 September 2014

Yang Zhang  and
Xiangui Zhao
Yang Zhang
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 email yang.zhang@umanitoba.ca
Xiangui Zhao
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 email xian.zhao@umanitoba.ca

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Differential difference algebras, introduced by Mansfield and Szanto, arose naturally from differential difference equations. In this paper, we investigate the Gelfand–Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand–Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand–Kirillov dimension under some specific conditions and construct an example to show that this upper bound cannot be sharpened any further.

MSC classification

Secondary: 16P90: Growth rate, Gelfand-Kirillov dimension 16S36: Ordinary and skew polynomial rings and semigroup rings
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 485 - 495
Copyright
© The Author(s) 2014 

References

Bell, A. D. and Goodearl, K. R., ‘Uniform rank over differential operator rings and Poincaré–Birkhoff–Witt extensions’, Pacific J. Math. 131 (1988) no. 1, 13–37.CrossRefGoogle Scholar
Courtois, N., Klimov, A., Patarin, J. and Shamir, A., ‘Efficient algorithms for solving overdefined systems of multivariate polynomial equations’, Advances in cryptology–EUROCRYPT 2000 (Springer, Berlin, 2000) 392–407.Google Scholar
Huh, C. and Kim, C. O., ‘Gelfand–Kirillov dimension of skew polynomial rings of automorphism type’, Comm. Algebra 24 (1996) no. 7, 2317–2323.CrossRefGoogle Scholar
Hydon, P. E., ‘Symmetries and first integrals of ordinary difference equations’, Proc. R. Soc. Lond. Ser. A 456 (2000) 2835–2855.Google Scholar
Kandri-Rody, A. and Weispfenning, V., ‘Non-commutative Gröbner bases in algebras of solvable type’, J. Symbolic Comput. 9 (1990) no. 1, 1–26.Google Scholar
Kassel, C., Quantum groups , Graduate Texts in Mathematics 155 (Springer, Berlin, 1995).CrossRefGoogle Scholar
Krause, G. and Lenagan, T., Growth of algebras and Gelfand–Kirillov dimension , Graduate Studies in Mathematics 22 (American Mathematical Society, Providence, RI, 2000).Google Scholar
Levandovskyy, V. and Schönemann, H., ‘Plural: a computer algebra system for noncommutative polynomial algebras’, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ACM Press, New York, 2003) 176–183.Google Scholar
Lorenz, M., ‘Gelfand-Kirillov dimension of skew polynomial rings’, J. Algebra 77 (1982) no. 1, 186–188.Google Scholar
Mansfield, E. L. and Szanto, A., ‘Elimination theory for differential difference polynomials’, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ACM Press, New York, 2003) 191–198.Google Scholar
Matczuk, J., ‘The Gelfand–Kirillov dimension of Poincaré–Birkhoff–Witt extensions’, Perspectives in rings theory (eds Van Oystaeyen, F. and Le Bruyn, L.; Kluwer Academic Publishers, Dordrecht, 1988) 221–226.Google Scholar
McConnell, J. C., Robson, J. C. and Small, L. W., Noncommutative Noetherian rings , Graduate Studies in Mathematics 30 (American Mathematical Society, Providence, RI, 2001).Google Scholar
Zhang, J. J., ‘A note on GK dimension of skew polynomial extensions’, Proc. Amer. Math. Soc. 125 (1997) no. 2, 363–374.CrossRefGoogle Scholar