On the structure of $P(n)_\ast P((n))$ for $p=2$

Abstract:

We show that $P(n)_\ast (P(n))$ for $p=2$ with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation $\epsilon$ nor the coproduct $\Delta$ are multiplicative. As a consequence the algebra structure of $P(n)_\ast (P(n))$ is slightly different from what was supposed to be the case. We give formulas for $\epsilon (xy)$ and $\Delta (xy)$ and show that the inversion of the formal group of $P(n)$ is induced by an antimultiplicative involution $\Xi :P(n)\rightarrow P(n)$. Some consequences for multiplicative and antimultiplicative automorphisms of $K(n)$ for $p=2$ are also discussed.