Mixable shuffles, quasi-shuffles and Hopf algebras.
The shuffle product, the quasi-shuffle product, the mixable shuffle product and the Rota-Baxter algebras have arisen and been studied in various contexts, but the relations between them have been mostly overlooked. This paper studies and clarifies some important relations between them. A Rota-Baxter algebra of weight \(a\) over a commutative ring \(k\) (\(a\) is in \(k\)) is a \(k\)-algebra \(R\) and a \(k\)-linear map \(P\) from \(R\) to \(R\) satisfying \(P(x)P(y)=P(xP(y))+P(P(x)y)+aP(xy)\). The shuffle product is well-known and gives a Rota-Baxter algebra of weight 0. Mixable shuffles arose in the study of Rota-Baxter algebras, especially free Rota-Baxter algebras. The quasi-shuffle product was given by \textit{M. E. Hoffman} [J. Algebr. Comb. 11, No. 1, 49-68 (2000; Zbl 0959.16021)] in a recursive manner. We do not give the technical definitions of these products here. The authors of the paper under review show first that the recursive definition of the quasi-shuffle product has an explicit form in terms of the mixable shuffle product, and that both can be derived from a Rota-Baxter algebra of weight 1. They are both generalizations of the shuffle product. They show that the quasi-shuffle product on a locally finite set is a subalgebra of a mixable shuffle algebra. Finally, they use the Hopf algebra structures on quasi-shuffle algebras to obtain Hopf algebra structures on a free Rota-Baxter algebra, which generalizes previous results of \textit{G. E. Andrews, L. Guo, W. Keigher} and \textit{K. Ono} [Trans. Am. Math. Soc. 355, No. 11, 4639-4656 (2003; Zbl 1056.16025)].
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