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NONEXISTENCE OF REFLEXIVE IDEALS IN IWASAWA ALGEBRAS OF CHEVALLEY TYPE
 

Summary: NONEXISTENCE OF REFLEXIVE IDEALS IN
IWASAWA ALGEBRAS OF CHEVALLEY TYPE
K. ARDAKOV, F. WEI AND J. J. ZHANG
Abstract. Let be a root system and let (Zp) be the standard Chevalley
Zp-Lie algebra associated to . For any integer t 1, let G be the uniform
pro-p group corresponding to the powerful Lie algebra pt(Zp) and suppose
that p 5. Then the Iwasawa algebra G has no nontrivial two-sided reflexive
ideals. This was previously proved by the authors for the root system A1.
0. Introduction
0.1. Prime ideals in Iwasawa algebras. One of the main projects in the study
of noncommutative Iwasawa algebras aims to understand the structure of two-sided
ideals in Iwasawa algebras G and G for compact p-adic analytic groups G. A list
of open questions in this project was posted in a survey paper by the first author
and Brown [AB]. Motivated by its connection to the Iwasawa theory of elliptic
curves in arithmetic geometry it is particularly interesting to understand the prime
ideals of G when G is an open subgroup of GL2(Zp). A reduction [A] shows that
this amounts to understanding the prime ideals of G when G is an open subgroup
of SL2(Zp). In a recent paper we introduced some machinery which allowed us to
determine every prime ideal of G for any open torsionfree subgroup G of SL2(Zp),
see [AWZ, Theorem C]. In this paper the theory developed in [AWZ] will be used

  

Source: Ardakov, Konstantin - School of Mathematical Sciences, University of Nottingham

 

Collections: Mathematics