Coloured peak algebras and Hopf algebras

Abstract

For G a finite abelian group, we study the properties of general equivalence relations on G n = G n ⋊ G-fraktur signn, the wreath product of G with the symmetric group G-fraktur signn, also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of script K sign Gn as well as graded connected Hopf subalgebras of ⊕n≤o script K sign G n. In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects.