Skip to main content

Hopf Algebras of Linear Recurring Sequences over Rings and Modules

Abstract

The module of linear recurring sequences over a commutative ring R can be considered as a Hopf algebra dual to the polynomial Hopf algebra over R. Under this approach, some notions and operations from the Hopf algebra theory have an interesting interpretation in terms of linear recurring sequences. Generalizations are also considered: linear recurring bisequences, sequences over modules, and k-sequences.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 1.

    V. A. Artamonov, “Structure of Hopf algebras,” in: Itogi Nauki i Tekhniki. Algebra, geometry, topology [in Russian], VINITI publ. (1991), 29, 3–63.

    Google Scholar 

  2. 2.

    Ju. A. Bakhturin, Basic Structures of Modern Algebra [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  3. 3.

    V. L. Kurakin, “Convolution of linear recurring sequences,” Russian Math. Surveys, 48, No.4, 249–250 (1993).

    Google Scholar 

  4. 4.

    V. L. Kurakin, “Structure of the Hopf algebras of linear recurring sequences,” Russian Math. Surveys, 48, No.5 (1993).

    Google Scholar 

  5. 5.

    V. L. Kurakin, “Hopf algebras of linear recurring sequences over commutative rings,” in: Proceedings of 2 Math. Conf. of Moscow State Social Univ., Nahabino, 1994 pp. 67–69.

  6. 6.

    V. L. Kurakin, “Hopf algebras dual to a polynomial algebra over a commutative ring,” to be published.

  7. 7.

    J. Y. Abuhlail, J. Gomez-Torrecillas, and R. Wisbauer, “Dual coalgebras of algebras over commutative rings,” J. Pure Appl. Algebra, 153, 107–120 (2000).

    Article  Google Scholar 

  8. 8.

    L. Cerlienco and F. Piras, “On the continuous dual of a polynomial bialgebra,” Comm. Algebra, 19, No.10, 2707–2727 (1991).

    Google Scholar 

  9. 9.

    W. Chin and J. Goldman, “Bialgebras of linear recursive sequences,” Comm. Algebra, 21, No.11, 3935–3952 (1993).

    Google Scholar 

  10. 10.

    P. Haukkanen, “On a convolution of linear recurring sequences over finite fields. I, II,” J. Algebra, 149, No.1, 179–182 (1992); 164 No. 2, 542–544 (1994).

    Article  Google Scholar 

  11. 11.

    V. L. Kurakin, A. S. Kuzmin, A. V. Mikhalev, and A. A. Nechaev, “Linear recurring sequences over rings and modules,” J. Math. Sci., 76, No.6, 2793–2915 (1995).

    Google Scholar 

  12. 12.

    V. L. Kurakin, A. V. Mikhalev, A. A. Nechaev, and V. N. Tsypyschev, “Linear and polylinear recurring sequences over abelian groups and modules,” J. Math. Sci., 102, No.6, 4598–4627 (2000).

    Article  Google Scholar 

  13. 13.

    B. Peterson and E. Y. Taft, “The Hopf algebra of linear recursive sequences,” Aequat. Math., 20, 1–17 (1980).

    Google Scholar 

  14. 14.

    E. Snapper, “Completely primary rings, I,” Ann. Math., 52, No.3, 666–693 (1950).

    Google Scholar 

  15. 15.

    M. F. Sweedler, Hopf Algebras, Benjamin, New York (1969).

    Google Scholar 

  16. 16.

    N. Zierler and W. H. Mills, “Products of linear recurring sequences,” J. Algebra, 27, No.1, 147–157 (1973).

    Article  Google Scholar 

Download references

Authors

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 113–148, 2003.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kurakin, V.L. Hopf Algebras of Linear Recurring Sequences over Rings and Modules. J Math Sci 128, 3402–3427 (2005). https://doi.org/10.1007/s10958-005-0279-8

Download citation

Keywords

  • Hopf Algebra
  • Commutative Ring
  • Algebra Theory
  • Interesting Interpretation