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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	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 + References 1. Levchuk, V. M. Generating Sets of Root Elements of Chevalley Groups Over a Field, Algebra Logika, 1983, vol. 22, no. 5, pp. 504-517 (in Russian). 2. Levchuk V. M. Parabolic Subgroups of Certain ABA-groups, Mathematical Notes, 1982, vol. 31, no. 4, pp. 259-267. DOI: 10.1007/BF01138934. 3. Koibaev, V. A. Elementary Nets in Linear Groups, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2011, vol. 17, no. 4, pp. 134-141 (in Russian). 4. Kuklina, S. K., Likhacheva, A. O. and Nuzhin, Ya. N. On Closedness of Carpets of Lie Type Over Commutative Rings, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, vol. 21, no. 3, pp. 192-196 (in Russian). 5. Carter, R. W. Finite Groups of Lie Type, London, Wiley and Sons, 1985, 556 p. 6. Koibaev, V. A., Kuklina, S. K., Likhacheva, A. O. and Nuzhin, Ya. N. Subgroups, of Chevalley Groups over a Locally Finite Field, Defined by a Family of Additive Subgroups, Mathematical Notes, 2017, vol. 102, no. 6, pp. 792-798. DOI: 10.1134/S0001434617110190. 7. Koibaev, V. A. and Nuzhin, Ya. N. \(k\)-Invariant Nets Over an Algebraic Extension of the Field \(k\), Siberian Mathematical Journal, 2017, vol. 58, no. 1, pp. 109-112. DOI: 10.1134/S0037446617010141. 8. Nuzhin, Ya. N. Intermediate Subgroups in the Chevalley Groups of Type \(B_l\), \(C_l\), \(F_4\), and \(G_2\) over the nonperfect fields of characteristic 2 and 3, Siberian Mathematical Journal, 2013, vol. 54, no. 1, pp. 119-123. DOI: 10.1134/S0037446613010151. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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