TY - JOUR ID - 87523 TI - Duality theory of $p$-adic Hopf algebras JO - Categories and General Algebraic Structures with Applications JA - CGASA LA - en SN - 2345-5853 AU - Mihara, Tomoki AD - University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 Japan Y1 - 2021 PY - 2021 VL - 14 IS - 1 SP - 81 EP - 118 KW - Pontryagin duality KW - $p$-adic KW - Hopf DO - 10.29252/cgasa.14.1.81 N2 - 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 $2$. UR - https://cgasa.sbu.ac.ir/article_87523.html L1 - https://cgasa.sbu.ac.ir/article_87523_90bb198d291c498c8cd128ce4c24faad.pdf ER -