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Quasi-shuffle products. (English) Zbl 0959.16021
Let $$k$$ be a subfield of $$\mathbb{C}$$, $$A$$ a locally finite graded set, and let $$\mathcal A=k\langle A\rangle$$ be the graded noncommutative polynomial algebra. Given a commutative associative operation on $$A$$ that adds degrees, the author defines a commutative multiplication $$*$$ on $$\mathcal A$$, called the quasi-shuffle product. Moreover, he shows that there is a Hopf algebra structure $$(\mathcal A,*,\Delta)$$ and an isomorphism $$\exp$$ of the shuffle Hopf algebra $$(\mathcal A,\text{III},\Delta)$$ onto $$(\mathcal A,*,\Delta)$$ such that both the set $$L$$ of Lyndon words on $$A$$ and their images $$\exp(L)$$ freely generate the algebra $$(\mathcal A,*)$$. The graded dual of $$(\mathcal A,*,\Delta)$$ and a $$q$$-deformation $$*_q$$ are also studied, and several examples are discussed.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W50 Graded rings and modules (associative rings and algebras) 16S36 Ordinary and skew polynomial rings and semigroup rings 16S80 Deformations of associative rings 05E05 Symmetric functions and generalizations
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