Odd Riemannian symmetric spaces associated to four-forms

Abstract

To any four-form satisfying certain system of quadratic equations a Lie superalgebra ᒄ = ᒄ 0 + ᒄ 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.

Riemannian supergeometry

“…In particular, the existence of a scalar superproduct on V forces V 1 to be even-dimensional. Note that we allow ·, · 0 to be indefinite-see also [9].…”

“…It is to be mentioned that the non-linear theory developed in this paper already has some infinitesimal counterpart in the mathematical literature; for example, Cortés [9] defines a notion of infinitesimal pseudo-Riemannian symmetric superspace, and Serganova [25] lists up involutive automorphisms of the simple Lie superalgebras over R and C.…”

The classification problem for pseudo-Riemannian symmetric spaces

“…It appears that it is impossible to find all solutions of (10) in a straightforward way. In the recent paper [18] Cortés reconsiders the situation and is able to prove the following: Proposition 4.10 (Cortés [18], Thm. 10).…”

“…We shall discuss it at the end of the present section, see Theorem 4.4. First we want to give an overview on the results on hyper-Kähler symmetric triples obtained by Alekseevsky, Cortés, and the authors in [2], [18], and [37].…”

“…E.g., in the Alekseevsky-Cortés construction one has simply to replace the real structure τ induced by J by a real structure coming from real structures on both factors H and E. In other words, one can work from the beginning with R 2 ⊗ R E 0 instead of H ⊗ C E, where E 0 is a real symplectic vector space. Hypersymplectic triples with abelian holonomy can be classified as in Theorem 4.2, see [18,19,1]. In addition, there is a variant of Example 4.12 producing hypersymplectic symmetric triples with non-abelian holonomy (one has to replace H n by R 2 ⊗ R 2n , the parameter p disappears).…”