# zbMATH — the first resource for mathematics

Hopf algebras of linear recurring sequences. (English. Russian original) Zbl 1062.16045
Discrete Math. Appl. 14, No. 2, 115-152 (2004); translation from Diskretn. Mat. 16, No. 2, 7-43 (2004).
B. Peterson and the reviewer identified linearly recursive sequences over a field $$k$$ as the continuous dual $$k[x]^0$$ to the polynomial algebra $$k[x]$$ and studied the Hopf algebra structure on these sequences dual to the Hopf algebra $$k[x]$$ with $$x$$ primitive [Aequationes Math. 20, 1–17 (1980; Zbl 0434.16008)]. R. G. Larson and the reviewer studied the bialgebra structure on these sequences dual to the bialgebra structure on $$k[x]$$ with $$x$$ group-like, and also the Hopf algebra structure on $$k[x,x^{-1}]^0$$ [Isr. J. Math. 72, No. 1–2, 118–132 (1990; Zbl 0729.11007)]. The author of the article under review generalized this for $$k=R$$, a commutative ring, and, with some of his colleagues, to modules over $$R$$.
This is an expository paper on these topics, analogous Hopf algebra and bialgebra structures are described on $$R[x]^0$$ and $$R[x,x^{-1}]^0$$. A version is given in terms of rational generating functions and in terms of representative functions. Group-like elements, primitive elements and integrals are identified. Linearly recursive sequences over a module are discussed and described as Hopf modules over linearly recursive sequences over $$R$$. A final section generalizes to sequences in several variables. This generalizes the case of $$R=k$$ a field, as studied by the reviewer [Discrete Math. 139, No. 1–3, 393–397 (1995; Zbl 0824.11007)].
##### MSC:
 16T05 Hopf algebras and their applications 11B37 Recurrences
##### Citations:
Zbl 0434.16008; Zbl 0729.11007; Zbl 0824.11007
Full Text:
##### References:
 [1] Artamonov V. A., J. Math. Sci. 71 pp 2289– (1994) [2] Yu. A. Bakhturin, Basic Structures of Modern Algebra. Kluwer, Dordrecht, 1993. · Zbl 0707.00001 [3] M. M. Glukhov, V. P. Elizarov, and A. A. Nechaev, Algebra. Helios, Moscow, 2003 (in Russian). [4] V., Russian Math. Surv. 48 (1993) pp 249– (1993) [5] V., Russian Math. Surv. 48 (5) pp 177– (1993) [6] V., Nakhabino pp 67– (1994) [7] Kurakin V. L., Math. Notes 71 pp 617– (2002) [8] R. Lidl and H. Niederreiter. Finite Fields. Addison-Wesley, London, 1983. · Zbl 0554.12010 [9] Zierler N., J. Soc. Industr. Appl. Math. 7 pp 31– (1959) [10] E. Abe, Hopf Algebras. Cambridge Univ. Press, Cambridge, 1980. [11] Abuhlail J. Y., J. Pure and Appl. Algebra 153 pp 107– (2000) [12] Abuhlail J. Y., J. Algebra 240 pp 165– (2001) [13] Cerlienco L., Commun. Algebra 19 (1991) pp 2707– (1991) [14] Cerruti U., J. Algebra 175 pp 332– (1995) [15] Chin W., Commun. Algebra 21 pp 3935– (1993) [16] J. Algebra 149 pp 179– (1992) [17] Kurakin V. L., J. Math. Sci. 76 pp 2793– (1995) [18] Kurakin V. L., J. Math. Sci. 102 pp 4598– (2000) [19] Kurakin V. L., Commun. Algebra 29 pp 4079– (2001) [20] Larson R., Israel J. Math. 72 pp 118– (1990) [21] S. Montogomery, Hopf Algebras and their Actions on Rings. AMS, Providence, 1993. [22] Peterson B., Math. 20 pp 1– (1980) [23] Snapper E., Ann. Math. 52 pp 666– (1950) [24] M. F. Sweedler, Hopf Algebras. Benjamin, New York, 1969. [25] Lect. Notes Pure Appl. Math. 158 pp 299– (1994) [26] Taft E., Congr. Numerantium 107 pp 33– (1995) [27] Zierler N., J. Algebra 27 (1973) pp 147– (1973)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.