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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 47

Automophisms of Nil-Triangular Subrings of Algebras Chevalley Type 𝐺2 Over Integral Domain. I

Author(s)
Alyona V. Kazakova1

1Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract
Let 𝑁Φ(𝐾) be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring 𝐾 with the identity associated with a root system Φ. This paper studies the well-known problem of describing automorphisms of Lie algebras and rings 𝑁Φ(𝐾). Automorphisms of the Lie algebra 𝑁Φ(𝐾) under restrictions 𝐾 = 2𝐾 = 3𝐾 on ring 𝐾 are described by Y. Cao, D. Jiang, J. Wang (2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type 𝐴𝑛, a description of automorphisms of Lie rings 𝑁Φ(𝐾) over 𝐾 was obtained by V.M. Levchuk (1983). Automorphisms of the Lie ring 𝑁Φ(𝐾) are described by V.M. Levchuk (1990) for type 𝐷4 over 𝐾, and for other types by A.V. Litavrin (2017), excluding types 𝐺2 and 𝐹4. In this paper we describe automorphisms of a nil-triangular Lie ring of type 𝐺2 over an integrity domain 𝐾 under the following restrictions on 𝐾: either 𝐾 = 2𝐾 = 3𝐾, or 3𝐾 = 0. To study automorphisms, the upper and lower central series described in this paper, are essentially used.
About the Authors
Alyona V. Kazakova, Postgraduate, Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, alvkazakova@gmail.com
For citation
Kazakova A. V. Automophisms of Nil-Triangular Subrings of Algebras Chevalley Type 𝐺2 Over Integral Domain. I. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 47, pp. 93–106. (in Russian) https://doi.org/10.26516/1997-7670.2024.47.93
Keywords
Chevalley algebra, nil-triangular subalgebra, ring, automorphism, hypercentral automorphism
UDC
512.554
MSC
17D99
DOI
https://doi.org/10.26516/1997-7670.2024.47.93
References
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