Structure of Singular Superalgebras with 2-Dimensional Even Part and New Examples of Singular Superalgebras

It is proved that a singular superalgebra with a 2-dimensional even part is isomorphic to a superalgebra B2|3(phi, xi, psi). In particular, there do not exist infinite-dimensional simple singular superalgebras with a 2-dimensional even part. It is proved that if a singular superalgebra contains an odd left annihilator, then it contains a nondegenerate switch. Lastly, it is established that for any number N >= 5, except the numbers 6, 7, 8, 11, there exist singular superalgebras with a switch of dimension N. For the numbers N = 6, 7, 8, 11, there do not exist singular N -dimensional superalgebras with a switch.

Automated summary

It has been proven that a singular superalgebra with a 2-dimensional even part is isomorphic to a superalgebra B2|3(phi, xi, psi). Particularly, there are no infinite-dimensional simple singular superalgebras with a 2-dimensional even part. It has been shown that if a singular superalgebra contains an odd left annihilator, then it contains a nondegenerate switch. Lastly, it has been established that for any number N >= 5, except the numbers 6, 7, 8, 11, there exist singular superalgebras with a switch of dimension N. For the numbers N = 6, 7, 8, 11, there do not exist singular N-dimensional superalgebras with a switch.