On irreducible carpets of additive subgroups of type $F_4$
A. O. Likhachevaab a North Caukasus Center for Mathematical Research NOSU, 46 Vatutina St., Vladikavkaz 362025, Russia
b School of Mathematics and Computer Science SibFU,
79 Svobodny Ave., Krasnoyarsk 660041, Russia
Abstract:
The article describes irreducible carpets
$\mathfrak{A}=\{\mathfrak{A}_r:\ r\in \Phi\}$ of type
$F_4$ over the field
$K$, all of whose additive subgroups
$\mathfrak{A}_r$ are
$R$-modules, where
$K$ is an algebraic extension of the field
$R$. An interesting fact is that carpets which are parametrized by a pair of additive subgroups appear only in characteristic 2. Up to conjugation by a diagonal element from the corresponding Chevalley group, this pair of additive subgroups becomes fields, but they may be different. In addition, we establish that such carpets
$\mathfrak{A}$ are closed. Previously, V. M. Levchuk described irreducible Lie type carpets of rank greater than
$1$ over the field
$K$, at least one of whose additive subgroups is an
$R$-module, where
$K$ is an algebraic extension of the field
$R$, under the assumption that the characteristic of the field
$K$ is different from
$0$ and
$2$ for types
$B_l$,
$C_l$,
$F_4$, while for type
$G_2$ it is different from
$0$,
$2$, and
$3$ [1]. For these characteristics, up to conjugation by a diagonal element, all additive subgroups of such carpets coincide with one intermediate subfield between
$R$ and
$K$.
Key words:
Chevalley group, carpet of additive subgroups, carpet subgroup, commutative ring.
UDC:
512.54
MSC: 20G15 Received: 03.03.2022
DOI:
10.46698/i7746-0636-8062-u
Fulltext:
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