Lie superalgebras with some homogeneous structures

TL;DR

This paper extends classical symplectic double extension concepts to Lie superalgebras, providing inductive descriptions and examples of various homogeneous quadratic symplectic structures, enriching the understanding of their algebraic properties.

Contribution

It introduces generalized double extension methods for Lie superalgebras, broadening the classification and structural understanding of homogeneous quadratic symplectic superalgebras.

Findings

Existence of new homogeneous quadratic symplectic Lie superalgebras

Inductive descriptions via generalized double extensions

Relations between quadratic symplectic superalgebras and Manin superalgebras

Abstract

We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [2]. We use this concept to give an inductive description of nilpotent homogeneous-symplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than even-quadratic even-symplectic considered in [6]. We study the structures of even (resp. odd)-quadratic odd (resp. even)-symplectic Lie superalgebras and odd-quadratic odd-symplectic Lie superalgebras and we give its inductive descriptions in terms of quadratic generalized double extensions and odd quadratic generalized double extensions. This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [6]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superargebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [6].