Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions

In his work on P-partitions, Stembridge defined the algebra of peak functions Pi, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show that Pi is closed under coproduct, and therefore a Hopf algebra, and describe the kernel of the retraction. Billey and Haiman, in their work on Schubert polynomials, also defined a new class of quasi-symmetric functions --- shifted quasi-symmetric functions --- and we show that Pi is strictly contained in the linear span Xi of shifted quasi-symmetric functions. We show that Xi is a coalgebra, and compute the rank of the n-th graded component.