Double extensions of quasi-Frobenius Lie superalgebras with degenerate center
Sofiane Bouarroudj, Quentin Ehret
TL;DR
The paper develops symplectic double extensions of quasi-Frobenius Lie superalgebras with degenerate center, extending Fischer's method to the super setting by introducing two standard models governed by orthosymplectic and periplectic forms with a homogeneous symplectic form $\\omega$. It proves that every double extension is equivalent to one of these standard models, and provides a canonical construction using an abelian extension together with cohomological data to satisfy Jacobi and invariance requirements. The work integrates parity considerations, cohomology, and explicit equivalence results to generalize the classical theory to Lie superalgebras. It concludes with concrete low-dimensional examples that illustrate the method and highlight potential directions for future research.
Abstract
We develop the process of symplectic double extensions for Lie superalgebras with degenerate center. The construction is a superization of a recent work by Fischer, and generalize our previous work. We provide a standard model for such double extensions, where the symplectic form is either orthosymplectic or periplectic. Additionally, we show that every double extension is naturally equivalent to either of these two standard types of extensions. Several examples in low dimensions are given to illustrate the concept.