On the closedness of carpets of additive subgroups associated with a Chevalley group over a commutative ring

Yakov N. Nuzhin

Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract: Let $\mathfrak{A}=\{\mathfrak{A}_r\ |\ r\in \Phi\}$ be a carpet of additive subgroups of type $\Phi$ over an arbitrary commutative ring $K$. A sufficient condition for the carpet $\mathfrak{A}$ to be closed is established. As a corollary, we obtain a positive answer to question 19.63 from the Kourovka notebook and a confirmation of one conjecture by V. M. Levchuk, provided that the type of $\Phi$ is different from $C_l$, $l\geqslant 5$ when the characteristic of the ring $K$ is $0$ or $2m$ for some natural number $m>1$. Also, a partial answer to question 19.62 has been obtained.

Keywords: Lie algebra and ring, Chevalley group, commutative ring, carpet of additive subgroups, carpet subgroup.

UDC: 512.54

Received: 10.06.2023
Received in revised form: 31.07.2023
Accepted: 04.09.2023

Language: English