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.