Local automorphisms of complex solvable Lie algebras of maximal rank

AbstractThis paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism.Keywords: Solvable Lie algebra of maximal ranknilradicaltorusroot systemautomorphismlocal automorphismAMS Subject Classifications (2020): 17A3617B2217B3017B40 AcknowledgmentsThe authors are indebted to the referees for valuable comments.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThe first author was partially supported by the Ministry of Science and Higher Education of the Russian Federation [agreement no. 075–02–2023–914].

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