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.
G., Fundamentals of Hopf Algebras , Universitext 44, Springer International Publiching Switzerland, Springer Cham, Heidelberg New York Dordrecht London, xiv+150pp (2015)
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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.