Lattice automorphisms of Lie algebras (Q1071098)

Let \(\theta\) be a lattice isomorphism of the Lie algebra \(L\) onto \(L^*\). The main results are: (1) Every non-constant lattice homomorphism is a lattice isomorphism. (2) If \(L\) is not abelian then each lattice automorphism of \(L\) is induced by at most one algebra automorphism of \(L\). (3) Let \(L=A\times B\) where \(A\cong B\) and \(A\) is not abelian. If \(L^*=A^*\times B^*\) then \(A^*\cong B^*\). A subalgebra of \(L\) is called \(\ell\)-characteristic if it is invariant under all lattice automorphisms of \(L\). (4) If \(L\) is of characteristic zero and has no proper \(\ell\)-characteristic subalgebras then either \(L\) is abelian, or almost abelian, or \(L=S_ 1\times\cdots\times S_ n\) where \(S_ i\)'s are lattice isomorphic simple Lie algebras. Remark. The first statement was proved by \textit{A. A. Lashkhi} in ``\({\mathcal L}\)-homomorphisms of Lie algebras'' [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 70, 64--68 (1981; Zbl 0525.17003)]. The second statement was proved by \textit{A. A. Lashkhi} in ``Lattice isomorphisms of some classes of Lie algebras'' [Tr. Tbilis. Mat. Inst. Razmadze 46, 5--21 (1975; Zbl 0335.17003)]. In the paper ``Minimal noncommutative and minimal nonabelian algebras'' [Commun. Algebra 13, 305--328 (1985; Zbl 0558.17008)] the reviewer gives the answer to the question of this paper: there exist lattice isomorphic simple algebras of arbitrary large dimension.