Non-existence of non-trivial normal elements in the Iwasawa Algebra of Chevalley groups

2019/07/31 by Dong Han, Han, Dong, Jishnu Ray +3
Mathematics · #11R23 #17B22 (Primary) #17B45 #22E35 #22E50 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1907.13333

openalex publication_date 2019/07/31 · openalex created_date 2019/08/13 · openalex updated_date 2026/07/28

Abstract

For a prime p>2, let G be a semi-simple, simply connected, split Chevalley group over ℤ_p, G(1) be the first congruence kernel of G and Ω_G(1) be the mod-p Iwasawa algebra defined over the finite field \mathbbF_p. Ardakov, Wei, Zhang have shown that if p is a "nice prime " (p ≥ 5 and p \nmid n+1 if the Lie algebra of G(1) is of type A_n), then every non-zero normal element in Ω_G(1) is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov, Wei and Zhang's result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture.

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