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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
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