Projections of semisimple Lie algebras
type: Item
instance of: scholarly article
title: Projections of semisimple Lie algebras
zbMATH Open document ID: 1475.17017
published in: Algebra and Logic
publication date: 2019-12-06T00:00:00Z
review text: The paper proves that any finite-dimensional Lie algebra over a perfect field of finite characteristic \(p>3\), different from a 3-dimensional simple non-split Lie algebra, that is lattice isomorphic to a semisimple Lie algebra is also semisimple. Here by lattice isomorphism of Lie algebras the author means the isomorphism of their subalgebra lattices. This result extends a similar result for Lie algebras over a field of characteristic \(0\) proved by the author in [\textit{A. G. Geĭn}, Izv. Vyssh. Uchebn. Zaved., Mat. 1978, No. 4(191), 26--31 (1978; Zbl 0403.17003)]. The main technical element of the proof is the analysis of modular subalgebras. Namely, an element \(M\) is called modular in a lattice \(\mathcal{L}\) iff a mapping \(X\to X \cap M\) is an isomorphism of an interval \([M; A\lor M]\) onto an interval \([A\cap M; A]\) for any element \(A\) in \(\mathcal{L}\). A subalgebra \(A\) is called modular, if it forms a modular elements in the subalgebra lattice. The proof is split into a number of lemmas analysing the cases of abelian and almost abelian modular subalgebras, modular subalgebras isomorphic to a 3-dimensional simple non-split Lie algebra and a special case of a subalgebra lattice of length 3 containing a three-dimensional simple non-split Lie algebra.
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 7140974
zbMATH Keywords: subalgebra lattice
zbMATH Keywords: lattice isomorphism
zbMATH Keywords: semisimple Lie algebras
zbMATH Keywords: modular subalgebra
Wikidata QID: Q115384411
author: A. G. Gein
reviewed by: Boris M. Dubrov
MaRDI profile type: Publication
full work available at URL: s10469-019-09529-z
OpenAlex ID: W2963694777
cites work: Lattic isomorphisms of lie algebras
cites work: Lattices of subalgebras of real Lie algebras
cites work: Q4187954
cites work: Q4165427
cites work: Modularity in Lie algebras
cites work: Q3820754
cites work: Q5732712
cites work: Q3810808
cites work: Lie algebras whose proper subalgebras are either semisimple, abelian or almost-abelian
cites work: Finite-dimensional simple Lie algebras with subalgebra lattice of length 3
cites work: Q3342713
DOI: 10.1007/S10469-019-09529-Z
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description: scientific article; zbMATH DE number 7140974