Geodesic
On irreducible carpets of additive subgroups of type $F_4$
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 2, pp. 117-123 
A. O. Likhacheva. On irreducible carpets of additive subgroups of type $F_4$. Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 2, pp. 117-123. http://geodesic.mathdoc.fr/item/VMJ_2023_25_2_a9/
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     title = {On irreducible carpets of additive subgroups of type $F_4$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $K$.

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[7] Koibaev V. A., Nuzhin Ya. N., “$k$-Invariant Nets Over an Algebraic Extension of the Field $k$”, Siberian Mathematical Journal, 58:1 (2017), 109–112 | DOI | DOI | MR | Zbl

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