Quasi-shuffle products
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.
scientific article