Shifted combinatorial Hopf algebras from K-theory

In prior joint work with Lewis, we developed a theory of enriched set-valued P-partitions to construct a K-theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of K-theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new K-theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse's K-theoretic Schur P- and Q-functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures.