Projections of semisimple Lie algebras
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.
scientific article; zbMATH DE number 7140974
scientific article