New directions in the study of prime ideals in rational, nilpotent Iwasawa algebras

TL;DR

This paper investigates the structure of prime ideals in Iwasawa algebras of nilpotent p-valuable groups, confirming a conjecture for several new classes of such groups.

Contribution

It proves the conjecture that prime ideals have a canonical form for new classes of nilpotent groups, expanding understanding of Iwasawa algebra structures.

Findings

Confirmed the canonical form conjecture for multiple new classes of nilpotent groups.

Excluded type C groups from the classes where the conjecture is proved.

Extended the classification of prime ideals in Iwasawa algebras.

Abstract

Let G be a nilpotent p-valuable (compact p-adic Lie) group. There is an ongoing investigation into the prime ideals of its completed group algebra (Iwasawa algebra), and there remains an open conjecture that they can all be proved to have a canonical standard form. We very this conjecture for several new classes of nilpotent groups, including those corresponding to the positive subalgebra of almost all classical and exceptional types, curiously excluding those of type C.