The Hopf algebra ${\cal C}( {\rm I\! F}_q[[T]] , K); \quad {\rm I\!F}_q((T)) \subseteq K$

Bertin DIARRA

Let G be a compact group that is totally disconnected. If K is a complete ultrametric valued field, it is well known that the Banach algebra ${\cal C}(G , K)$ of the continuous functions of G with values in K is a complete ultrametric Hopf algebra with coproduct induced by the multiplication of G. We are concerned here with the case when $G = {\rm I\! F}_q[[T]]$ is the additive group of the ring of formal power series with coefficients in the finite field ${\rm I\! F}_q$ and K is a valued field whose valuation extends the T-adic valuation of the field of formal Laurent series ${\rm I\!F}_q((T))$. Thank to Carlitz-Wagner orthonormal basis of ${\cal C}( {\rm I\! F}_q[[T]] , K)$, the Hopf algebra ${\cal C}( {\rm I\! F}_q[[T]] , K)$ is seen to be a binomial divided power coalgebra.

We give a description of the continuous bialgebra endomorphisms of ${\cal C}( {\rm I\! F}_q[[T]] , K)$ and that of the continuous comodule endomorphisms of the same coalgebra, considered as a comodule over itself.