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Lagrangian extensions and left symmetric structures on the four-dimensional real Lie superalgebras

Sofiane Bouarroudj, Ana-Maria Radu

Abstract

Over real numbers, Backhouse classified all four-dimensional Lie superalgebras. From this list, we will investigate those Lie superalgebras that can be obtained as Lagrangian extensions. Moreover, we investigate left-symmetric structures on these Lie superalgebras. Furthermore, except for two of them, they are all Novikov superalgebras.

Lagrangian extensions and left symmetric structures on the four-dimensional real Lie superalgebras

Abstract

Over real numbers, Backhouse classified all four-dimensional Lie superalgebras. From this list, we will investigate those Lie superalgebras that can be obtained as Lagrangian extensions. Moreover, we investigate left-symmetric structures on these Lie superalgebras. Furthermore, except for two of them, they are all Novikov superalgebras.
Paper Structure (17 sections, 5 theorems, 24 equations, 13 tables)

This paper contains 17 sections, 5 theorems, 24 equations, 13 tables.

Table of Contents

  1. Introduction
  2. Lagrangian extensions of Lie (super)algebras
  3. Left-symmetric structures
  4. Main concepts and definitions
  5. Connections on Lie (super)algebras
  6. Left-symmetric (super)algebras
  7. The 4-dimensional real Lie superalgebras
  8. The notion of T*- extensions and Pi T*- extensions
  9. Flat connections and representations
  10. Polarization and Lagrangian extensions
  11. The 4-dimensional Lie superalgebras as Lagrangian extensions
  12. Left-symmetric structures on the 4-dimensional real Lie superalgebras
  13. The list of four-dimensional real Lie superalgebras
  14. Appendix: The decomposable case
  15. Abelian LSA
  16. ...and 2 more sections

Key Result

Proposition 2.1

If $V=V_{\bar{0}} \oplus V_{\bar{1}}$ is a finite-dimensional superspace such that $V_{\bar{1}} \not = {0}$, equipped with a non-degenerate anti-symmetric bilinear form $\omega$, then:

Theorems & Definitions (8)

  • Proposition 2.1: Superdimension constraints
  • proof
  • Lemma 3.1: Dual representation
  • proof
  • Lemma 3.2: Conditions on $\alpha$ and $\beta$
  • proof
  • Theorem 3.3: Lagrangian or $T^*$- and $\Pi T^*$-extensions
  • Theorem 3.4: Converse of Theorem \ref{['Tstar']}