Explicit antipode formulas are proved for commutative and non-commutative q-quasi-symmetric functions, plus a partial antipode on a new fundamental basis of NCQSym, via a cancelation argument that recovers the classical QSym antipode.
Sch\"utzenberger's factorization on the (completed) Hopf algebra of $q-$stuffle product
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abstract
In order to extend the Sch\"utzenberger's factorization, the combinatorial Hopf algebra of the $q$-stuffles product is developed systematically in a parallel way with that of the shuffle product and and in emphasizing the Lie elements as studied by Ree. In particular, we will give here an effective construction of pair of bases in duality.
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The Antipodes of $q$-Quasi-Symmetric Functions and Non-Commutative Quasi-Symmetric Functions
Explicit antipode formulas are proved for commutative and non-commutative q-quasi-symmetric functions, plus a partial antipode on a new fundamental basis of NCQSym, via a cancelation argument that recovers the classical QSym antipode.