%0 Journal Article
%T Duality theory of $p$-adic Hopf algebras
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Mihara, Tomoki
%D 2021
%\ 01/01/2021
%V 14
%N 1
%P 81-118
%! Duality theory of $p$-adic Hopf algebras
%K Pontryagin duality
%K $p$-adic
%K Hopf
%R 10.29252/cgasa.14.1.81
%X We show the monoidal functoriality of Schikhof duality, and cultivate new duality theory of $p$-adic Hopf algebras. Through the duality, we introduce two sorts of $p$-adic Pontryagin dualities. One is a duality between discrete Abelian groups and affine formal group schemes of specific type, and the other one is a duality between profinite Abelian groups and analytic groups of specific type. We extend Amice transform to a $p$-adic Fourier transform compatible with the second $p$-adic Pontryagin duality. As applications, we give explicit presentations of a universal family of irreducible $p$-adic unitary Banach representations of the open unit disc of the general linear group and its $q$-deformation in the case of dimension $2$.
%U https://cgasa.sbu.ac.ir/article_87523_90bb198d291c498c8cd128ce4c24faad.pdf