On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras

Let $\mathcal G\simeq H\rtimesΓ$ be the semidirect product of a finite group $H$ and $Γ\simeq\mathbb Z_p$. Let $F/\mathbb Q_p$ be a finite extension with ring of integers $\mathcal O_F$. Then the total ring of quotients $\mathcal Q^F(\mathcal G)$ of the completed group ring $\mathcal O_{F}[[\mathcal G]]$ is a semisimple ring. We determine its Wedderburn decomposition under a ramification hypothesis by relating it to the Wedderburn decomposition of the group ring $F[H]$.

💡 Research Summary

The paper investigates the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras associated with a profinite group G that splits as a semidirect product G = H ⋊ Γ, where H is a finite group and Γ ≅ ℤₚ is a one‑dimensional p‑adic Lie group. For a finite extension F/ℚₚ with ring of integers 𝒪_F, the completed group algebra Λ_{𝒪_F}(G) = 𝒪_F