Simple finite-dimensional binary Lie superalgebras

Abstract

We prove that every complex finite-dimensional simple binary Lie superalgebra with non-zero odd part is either a simple Lie superalgebra, and hence is one of the superalgebras of Kac's list, or the two-dimensional non-Lie superalgebra $B_2$. This confirms the Grishkov--Shestakov conjecture and, together with the theorems of Grishkov and Filippov on simple binary Lie algebras, gives the answer to Shestakov's question from the Dniester Notebook (Problem 3.100(d)) over the field of complex numbers.