Intersection of ideals with non-commutative subalgebras

ABSTRACT

Computation of an intersection of a left ideal with a subalgebra, which is not fully investigated until now, is important for different areas of mathematics.We present an algorithm for the computation of the preimage of a left ideal under a morphism of non-commutative GR-algebras, and show both its abilities and limitations.The main computational tools are the elimination of variables by means of Gröbner bases together with the constructive treatment of opposite algebras and the utilization of a special bimodule structure.

REFERENCES

• 1: Apel, J. Gröbnerbasen in nichtkommutativen Algebren und ihre Anwendung. Dissertation, Universität Leipzig, 1988.
• 2: Borges, M. A. and Borges, M. Gröbner bases property on elimination ideal in the noncommutative case. In B. Buchberger and F. Winkler, editors, Gröbner bases and applications, pages 323--337. Cambridge University Press, 1998.
• 3: Bueso, J., Gómez-Torrecillas, J. and Verschoren, A. Algorithmic methods in non-commutative algebra. Applications to quantum groups. Kluwer Academic Publishers, 2003.
• 4: Frédéric Chyzak , Bruno Salvy, Non-commmutative elimination in ore algebras proves multivariate identities, Journal of Symbolic Computation, v.26 n.2, p.187-227, Aug. 1998 [doi>10.1006/jsco.1998.0207]
• 5: García Román, M. and García Román, S. Gröbner bases and syzygies on bimodules over PBW algebras. Journal of Symbolic Computation, 40(3):1039--1052, 2005.
• 6: Gómez-Torrecillas, J. and Lobillo, F.J. Auslander-regular and Cohen-Macaulay quantum groups. J. Algebr. Represent. Theory, 7(1):35--42, 2004.
• 7: Edward L. Green, Multiplicative bases, Gröbner bases, and right Gröbner bases, Journal of Symbolic Computation, v.29 n.4-5, p.601-623, April/May 2000 [doi>10.1006/jsco.1999.0324]
• 8: Greuel, G.-M. and Pfister, G. with contributions by Bachmann, O., Lossen, C. and Schönemann, H. A SINGULAR Introduction to Commutative Algebra. Springer, 2002.
• 9: Greuel, G.-M., Pfister G., and Schönemann H. Singular 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, 2005. Available from http://www.singular.uni-kl.de.
• 10: Greuel, G.-M., Levandovskyy, V. and Schönemann H. Plural. A Subsystem of the Computer Algebra System Singular for Computations with Non-commutative Polynomial Algebras. Centre for Computer Algebra, University of Kaiserslautern, 2005. Available from http://www.singular.uni-kl.de.
• 11: A. Kandr-Rody , V. Weispfenning, Non-commutative Gro¨bner bases in algebras of solvable type, Journal of Symbolic Computation, v.9 n.1, p.1-26, Jan. 1990 [doi>10.1016/S0747-7171(08)80003-X]
• 12: Kredel, H. Solvable polynomial rings. Shaker, 1993.
• 13: Levandovskyy, V. Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation. Doctoral Thesis, Universität Kaiserslautern, 2005. Available from http://kluedo.uni-kl.de/volltexte/2005/1883/.
• 14: Levandovskyy, V. On preimages of ideals in certain non-commutative algebras. In Pfister G., Cojocaru S. and Ufnarovski, V., editor, Computational Commutative and Non-Commutative Algebraic Geometry. IOS Press, 2005.
• 15: Levandovskyy, V. PBW Bases, Non-Degeneracy Conditions and Applications. In Buchweitz, R.-O. and Lenzing, H., editor, Representation of algebras and related topics. Proceedings of the ICRA X conference, volume 45, pages 229--246. AMS. Fields Institute Communications, 2005.
• 16: Viktor Levandovskyy , Hans Schönemann, Plural: a computer algebra system for noncommutative polynomial algebras, Proceedings of the 2003 international symposium on Symbolic and algebraic computation, p.176-183, August 03-06, 2003, Philadelphia, PA, USA [doi>10.1145/860854.860895]
• 17: Levandovskyy, V., Lobillo, F.J. and Rabelo, C. A Singular 3.0 library, providing general tools for noncommutative algebras nctools.lib. 2004.
• 18: Li, H. Noncommutative Gröbner bases and filtered-graded transfer. Springer, 2002.
• 19: McConnell, J.C. and Robson, J.C. Noncommutative Noetherian rings. With the cooperation of L. W. Small. Graduate Studies in Mathematics. 30. Providence, RI: American Mathematical Society (AMS), 2001.
• 20: Teo Mora, Groebner Bases in Non-Commutative Algebras, Proceedings of the International Symposium ISSAC'88 on Symbolic and Algebraic Computation, p.150-161, July 04-08, 1988
• 21: Nordbeck, P. On some basic applications of Gröbner bases in non-commutative polynomial rings. In B. Buchberger and F. Winkler, editors, Gröbner bases and applications, pages 463--472. Cambridge University Press, 1998.
• 22: I. Shestakov and U. Umirbaev. Free Akivis algebras, primitive elements, and hyperalgebras. J. Algebra, 250(2):533--548, 2002.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

Additional Classification:
  H. Information Systems
  H.4 INFORMATION SYSTEMS APPLICATIONS

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

General Terms:
Algorithms, Design, Theory

Keywords:
Groebner bases, elimination, homomorphism of algebras, intersection with subalgebra, non-commutative algebra, preimage of ideal, restriction