Simple binary Lie and non-Lie superalgebra has solvable even part
Binary Lie algebras were introduced by \textit{A. I. Mal'tsev} [Mat. Sb., Nov. Ser. 36(78), 569--576 (1955; Zbl 0065.00702)] as anticommutative algebras in which any two elements generate a Lie subalgebra. Later, \textit{A. T. Gaĭnov} [Usp. Mat. Nauk 12, No. 3(75), 141--146 (1957; Zbl 0079.04801)] characterized binary Lie algebras by identities: an anticommutative algebra is a binary Lie algebra if and only if it satisfies the identity \(J(xy, x, y)=0.\) \textit{A. N. Grishkov} [Math. USSR, Izv. 17, 243--269 (1981; Zbl 0468.17007); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 999--1030 (1980)] proved that every simple finite-dimensional binary Lie algebra over a field of characteristic \(0\) is a Malcev algebra, that is, is a Lie algebra or is isomorphic to the \(7\)-dimensional algebra of octonions with zero trace under the product defined by the commutator. On the other hand, the classification of complex simple finite-dimensional Lie superalgebras was given by Kac in 1977. The authors of the present paper continue their recent work about the classification of simple binary Lie superalgebras. Namely, they are working under the following conjecture.\N\NConjecture. Let \(B = B_0 \oplus B_1\) be a nontrivial complex finite-dimensional simple binary Lie superalgebra. Then \(B\) is a simple Lie superalgebra or \(\dim B = 2.\)\N\NThe following strategy for proving the conjecture is presented:\N\N1. Reduction to the case when \(B_0\) is solvable.\N\N2. Reduction to the case when \(B_0\) is nilpotent.\N\N3. Reduction to the case when \(B_0\) is abelian.\N\N4. To prove Conjecture for abelian even part.\N\NIn this paper, the authors prove that if \(B\) is not a Lie superalgebra then the even part \(B_0\) of \(B\) is solvable. Hence they realize the first step of the above strategy.
scientific article; zbMATH DE number 7874676
scientific article