Odd Riemannian symmetric spaces associated to four-forms

Abstract

To any four-form $\Omega$ satisfying certain system of quadratic equations a Lie superalgebra $\mathfrak{g} = \mathfrak{g}_0 + \mathfrak{g}_1$ endowed with an invariant scalar superproduct is associated. The geometric structure of the corresponding odd symmetric superspace, which is of hyper-Kähler type, is described. Finally, some classes of solutions to the quadratic system are discussed.