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Decision of Algebra Isomorphisms Using Gröbner Bases

  • Kiyoshi Shirayanagi
Conference paper
Part of the Progress in Mathematics book series (PM, volume 109)

Abstract

Given two finite-dimensional non-commutative finitely presented algebras A and B over a field k, this paper presents two algorithms for deciding whether or not they are isomorphic as k-algebras. We reduce this problem to a radical ideal membership problem in a commutative ring, by Hilbert’s zero point theorem. That is, A and B are isomorphic if and only if the determinant f of a k-linear mapping: AB does not lie in the radical of an ideal I derived from the relations of A and B. Moreover an efficient technique for computing f is provided.

We propose two Gröbner basis methods for solving our problem. One is to compute the radical of I directly and then to rewrite the above determinant f modulo this radical. The other method is to judge solvability of a certain system of algebraic equations: the union of I and a new polynomial. As a result, it is shown that the isomorphism problem for finitely presented algebras is decidable.

Keywords

Commutative Ring Ring Homomorphism Isomorphism Problem Algebra Isomorphism Algebraic Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Alonso, M. E., Mora, T. and Raimondo, M., Local decomposition algorithms, Proc. AAECC 8, L. N. Comp. Sci. 508 (1991), 208–221.MathSciNetGoogle Scholar
  2. [2]
    Buchberger, B., Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, Chapter 6 in Multidimensional Systems Theory (N. K. Bose ed.), D. Reidel Publishing Company (1985), 184–232.Google Scholar
  3. [3]
    Gateva-Ivanova, T. and Latyshev, V., On recognisable properties of associative algebras, J. Symbolic Computation 6 (1988), 371–388.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    Gianni, P. and Mora, T., Algebraic solution of systems of polynomial equations using Groebner bases, Proc. AAECC 5, L. N. Comp. Sci. 356 (1989), 247–257.MathSciNetGoogle Scholar
  5. [5]
    Gianni, P., Trager B., and Zacharias G., Gröbner Bases and Primary Decomposition of Polynomial Ideals, J. Symbolic Computation 6 (1988), 149–168.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Glass, A. M. W., The Word and Isomorphism Problems in Universal Algebra, Proc. Universal Algebra and Lattice Theory, L. N. Math. 1149 (1984), 123–128.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Kandri-Rody, A. and Weispfenning, V., Non-commutative Gröbner Bases in Algebras of Solvable Type, J. Symbolic Computation 9 (1990), 1–26.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Krick, T. and Logar, A., An algorithm for the computation of the radical of an ideal in the ring of polynomials, Proc. AAECC 9, L. N. Comp. Sci. 539 (1991), 195–205.MathSciNetGoogle Scholar
  9. [9]
    Le Chenadec, P., Canonical Forms in Finitely Presented Algebras, Research Notes in Theoretical Computer Science, Pitman, (1986).Google Scholar
  10. [10]
    Mora, T., Groebner bases for non-commutative polynomial rings, Proc. AAECC 3, L. N. Comp. Sci. 229 (1986), 353–362.MathSciNetGoogle Scholar
  11. [11]
    Seidenberg, A., Constructions in algebra, Trans. Am. Math. Soc. 197 (1974), 273–313.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Shirayanagi, K., A Classification of Finite-dimensional Monomial Algebras, Effective Methods in Algebraic Geometry (T. Mora and C. Traverso eds.), Progress in Mathematics 94, Birkhäuser (1991), 469–482.Google Scholar
  13. [13]
    Shirayanagi, K., On the Isomorphism Problem for Finite-dimensional Binomial Algebras, Proc. International Symposium on Symbolic and Algebraic Computation (ISSAC-90) (1990), 106–111.Google Scholar
  14. [14]
    Ufnarovskij, V., A growth criterion for graphs and algebras defined by words, Mat. Zametki 31, 465–472 (in Russian); English transl., Math. Notes 31 (1982), 238–241.Google Scholar
  15. [15]
    Vasconcelos, W., Jacobian matrices and constructions in algebra, Proc. AAECC 9, L. N. Comp. Sci. 539 (1991), 48–64.MathSciNetGoogle Scholar
  16. [16]
    Zariski, O. and Samuel, P., Commutative Algebra Vol. II, Van Nostrand, (1960).Google Scholar

Copyright information

© Birkhäuser Boston 1993

Authors and Affiliations

  • Kiyoshi Shirayanagi
    • 1
  1. 1.Communication Science LaboratoriesNTTSoraku-gun, KyotoJapan

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