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

Adam Jones, William Woods

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.

New directions in the study of prime ideals in rational, nilpotent 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.

Paper Structure

This paper contains 14 sections, 29 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $p$-valuable groups
  4. Lazard's equivalence of categories
  5. Right ideals and controlling subgroups
  6. Control theorems
  7. Old Control Theorem: The subgroup $A(G)$
  8. New Control Theorem: The subgroup $B(G)$
  9. Lie theoretic interpretation
  10. Examples
  11. Positive root bases
  12. Positive root subalgebras
  13. Comparison to $A(G)$
  14. Obstacles to a complete proof

Key Result

Lemma 2.1

Let $G$ be a group, $n$ any positive integer, and $a,b,c\in G$ arbitrary elements.

Theorems & Definitions (66)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Example 2.9
  • Remark 2.10
  • Lemma 2.11
  • ...and 56 more