Conjugation invariants in the Leibniz-Hopf algebra

The Leibniz-Hopf algebra is the Hopf algebra whose underlying algebra is the free associative algebra over Z on generators S1,S2,S3,… and whose diagonal is given by Δ(Sn)=∑i=0nSi⊗Sn−i (where S0 is to be understood as 1). We calculate a basis for the free submodule formed by the fixed points of this algebra under the Hopf algebra conjugation operation, χ. We also give bases for the submodules of fixed points in the dual Hopf algebra and in the mod p reductions of both the Leibniz-Hopf algebra and its dual.