It is well-known that any associative algebra becomes a Lie algebra under the new product given by the commutator. Similarly, under the symmetrized product (\(x\cdot y=xy+yx\)), any antiassociative algebra \(A\) (that is, \((xy)z=-x(yz)\)) satisfies the Jacobi identity: \((x\cdot y)\cdot z +(y\cdot z)\cdot x + (z\cdot x)\cdot y =0\). Moreover, if the characteristic is not two, then \(A^4=0\), so that \((A,\cdot)\) is a nilpotent Jordan algebra.
The paper under review deals with the super versions of the facts mentioned above and does it in a unifying way. The Jordan-Lie superalgebras referred to in the title are the superalgebras satisfying the graded versions of commutativity and of the Jacobi identity.
A concept of Jordan-Lie triple system along the same lines is introduced too, as well as some color-Lie superalgebras graded over \({\mathbb Z}_2\times{\mathbb Z}_2\), and several examples are given.