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Extensions of pseudo-euclidean mock-Lie superalgebras

Tahar Benyoussef, Sami Mabrouk

TL;DR

The paper extends mock-Lie theory by introducing pseudo-euclidean structures and a structured extension program. It proves that mock-Lie superalgebras are Jordan superalgebras and develops aRepresentation framework (adjoint, dual, coadjoint) together with semidirect product constructions. It then builds invariant bilinear forms through $T^*$-extensions and analyzes when extensions preserve pseudo-riemannian metrics, culminating in double and generalized double extensions via admissible pairs $(D,x_0)$ and their isometries. The results provide an inductive, constructive scheme to obtain all finite-dimensional pseudo-euclidean mock-Lie algebras from simpler components, with potential links to quadratic Lie/Leibniz-type structures and open questions around anti-Leibniz analogues.

Abstract

In this article, we introduce mock-Lie superalgebras, we give some definitions, properties, constructions, and we study their representations. Moreover we introduce pseudo-euclidean mock-Lie superalgebras which are mock-Lie superalgebras with even non-degenerate supersymmetric and invariant bilinear forms. Finally, we study the double extensions and generalized double extensions of mock-Lie superalgebras and their isometries.

Extensions of pseudo-euclidean mock-Lie superalgebras

TL;DR

The paper extends mock-Lie theory by introducing pseudo-euclidean structures and a structured extension program. It proves that mock-Lie superalgebras are Jordan superalgebras and develops aRepresentation framework (adjoint, dual, coadjoint) together with semidirect product constructions. It then builds invariant bilinear forms through -extensions and analyzes when extensions preserve pseudo-riemannian metrics, culminating in double and generalized double extensions via admissible pairs and their isometries. The results provide an inductive, constructive scheme to obtain all finite-dimensional pseudo-euclidean mock-Lie algebras from simpler components, with potential links to quadratic Lie/Leibniz-type structures and open questions around anti-Leibniz analogues.

Abstract

In this article, we introduce mock-Lie superalgebras, we give some definitions, properties, constructions, and we study their representations. Moreover we introduce pseudo-euclidean mock-Lie superalgebras which are mock-Lie superalgebras with even non-degenerate supersymmetric and invariant bilinear forms. Finally, we study the double extensions and generalized double extensions of mock-Lie superalgebras and their isometries.
Paper Structure (8 sections, 26 theorems, 85 equations)

This paper contains 8 sections, 26 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Mock-Lie superalgebras
  3. Definition and constructions
  4. Representation of mock-Lie superalgebras
  5. Pseudo-euclidean mock-Lie superalgebras
  6. Extensions of pseudo-euclidean mock-Lie superalgebras
  7. Doubles extensions
  8. Generalized double extensions

Key Result

Proposition 2.1

Every mock-Lie superalgebra is a Jordan superalgebra verifying $x^3=0$.

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.3
  • Example 2.5
  • ...and 53 more