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We develop a theory of multigraded (i.e., $N^l$-graded) combinatorial Hopfalgebras modeled on the theory of graded combinatorial Hopf algebras developedby Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. Inparticular we introduce the notion of canonical $k$-odd and $k$-evensubalgebras associated with any multigraded combinatorial Hopf algebra,extending simultaneously the work of Aguiar et al. and Ehrenborg. Among ourresults are specific categorical results for higher level quasisymmetricfunctions, several basis change formulas, and a generalization of thedescents-to-peaks map.

Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras