Simple Right-Alternative Unital Superalgebras Over an Algebra of Matrices of Order 2

Abstract

We classify simple right-alternative unital superalgebras over a field of characteristic not 2, whose even part coincides with an algebra of matrices of order 2. It is proved that such a superalgebra either is a Wall double W_2|2(ω), or is a Shestakov superalgebra S_4|2(σ) (characteristic 3), or is isomorphic to an asymmetric double, an 8-dimensional superalgebra depending on four parameters. In the case of an algebraically closed base field, every such superalgebra is isomorphic to an associative Wall double M₂[√1], an alternative 6-dimensional Shestakov superalgebra B_4|2 (characteristic 3), or an 8-dimensional Silva–Murakami–Shestakov superalgebra.