Duality theory of $p$-adic Hopf algebras

• Main Author: Tomoki Mihara
• Format: Article
• Language: English
• Published: Shahid Beheshti University 2021-01-01
• Series: Categories and General Algebraic Structures with Applications
• Subjects: pontryagin duality; $p$-adic; hopf
• Online Access: https://cgasa.sbu.ac.ir/article_87523_90bb198d291c498c8cd128ce4c24faad.pdf
• ISSN: 2345-5853; 2345-5861

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 $2$.