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DOI: 10.46698/i7746-0636-8062-u

On Irreducible Carpets of Additive Subgroups of Type \(F_4\)

Likhacheva, A. O.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 2.
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 \(K\).
Keywords: Chevalley group, carpet of additive subgroups, carpet subgroup, commutative ring.
Language: Russian Download the full text  
For citation: Likhacheva, A. O. On Irreducible Carpets of Additive Subgroups of Type \(F_4\), Vladikavkaz Math. J., 2023, vol. 25, no. 2, pp. 117-123 (in Russian). DOI 10.46698/i7746-0636-8062-u
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