2011/09/06 by Foissy, Loïc
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1109.1101
We study the self-dual Hopf algebra \h_\SP of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from \h_\SP to to the Hopf algebra of free quasi-symmetric functions \FQSym given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to \FQSym; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomorphism between these two Hopf subalgebras is also defined with the help of two transformations on special posets. The restriction of the Hopf pairing of \h_\SP to these Hopf subalgebras and others is also studied, as well as certain isometries between them. These problems are solved using duplicial and dendriform structures.An error in Section 7 has been noticed by Darij Grinberg, and the text has been modified accordingly.