On q-symmetric functions and q-quasisymmetric functions

In this paper, we construct the q-analogue of Poirier-Reutenauer algebras, related deeply with other q-combinatorial Hopf algebras. As an application, we use them to realize the odd Schur functions defined by Ellis, Khovanov, and Lauda, then naturally obtain the odd Littlewood-Richardson rule concerned by Ellis. Moreover, we construct the refinement of the odd Schur functions, called odd quasisymmetric Schur functions, parallel to the consideration by Haglund, Luoto, Mason, and van Willigenburg. All the q-Hopf algebras we discuss here provide the corresponding q-dual graded graphs.