For a prime $$p>2$$ p > 2 , let G be a semi-simple, simply connected, split Chevalley group over $${\mathbb {Z}}_p$$ Z p , G (1) be the first congruence kernel of G and $$\Omega _{G(1)}$$ Ω G ( 1 ) be the mod- p Iwasawa algebra defined over the finite field $${\mathbb {F}}_p$$ F p . Ardakov et al. (Adv Math 218: 865–901, 2008) have shown that if p is a “nice prime ” ( $$p \ge 5$$ p ≥ 5 and $$p \not \mid (n+1)$$ p ∤ ( n + 1 ) if the Lie algebra of G (1) is of type $$A_n$$ A n ), then every non-zero normal element in $$\Omega _{G(1)}$$ 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 et al. 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. We also propose some potential topics regarding to the normal elements and ideals in the Iwasawa algebras of the pro- p Iwahori subgroups of general linear group $${\hbox {GL}}_n(\mathbb {Z}_p)$$ GL n ( Z p ) and discuss how to extend our current techniques and methods to the case of the pro- p Iwahori subgroups of $${\hbox {GL}}_n(\mathbb {Z}_p)$$ GL n ( Z p ) .

中文翻译：

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Chevalley群的Iwasawa代数中的正规元

对于素数 $$p>2$$ p > 2 ，令 G 是一个半简单的、单连通的、分裂的 Chevalley 群在 $${\mathbb {Z}}_p$$ Z p 上，G (1) 是G 和 $$\Omega _{G(1)}$$ Ω G ( 1 ) 的第一同余核是在有限域 $${\mathbb {F}}_p$$ F p 上定义的 mod-p Iwasawa 代数. 阿尔达科夫等人。(Adv Math 218: 865–901, 2008) 已经证明如果 p 是一个“好的质数” ( $$p \ge 5$$ p ≥ 5 and $$p \not \mid (n+1)$$ p ∤ ( n + 1 ) 如果 G (1) 的李代数的类型是 $$A_n$$ A n )，则 $$\Omega _{G(1)}$$ Ω G 中的每个非零法向元素(1)是一个单位。此外，他们在他们的论文中推测他们良好的素数条件是多余的。本文的主要目标是提供对 Ardakov 等人的全新证明。结果使用本文第二作者开发的岩泽代数的显式表示，从而消除了好的素数条件，从而证明了他们的猜想。我们还提出了一些关于一般线性群 $${\hbox {GL}}_n(\mathbb {Z}_p)$$ GL n 的 pro-p Iwahori 子群的 Iwasawa 代数中的正规元和理想的一些潜在主题( Z p ) 并讨论如何将我们当前的技术和方法扩展到 $${\hbox {GL}}_n(\mathbb {Z}_p)$$ GL n ( Z p )。