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Quadratic symplectic Lie superalgebras over filiform modules

Elisabete Barreiro, Saïd Benayadi, Rosa M. Navarro, José M. Sánchez

Abstract

The present work studies deeply quadratic symplectic Lie superalgebras, obtaining, in particular, that they are all nilpotent. Consequently, we provide classifications in low dimensions and identify the double extensions that maintain symplectic structures. By means of both elementary odd double extensions and generalized double extensions of quadratic symplectic Lie superalgebras, we obtain an inductive description of quadratic symplectic Lie superalgebras of filiform type.

Quadratic symplectic Lie superalgebras over filiform modules

Abstract

The present work studies deeply quadratic symplectic Lie superalgebras, obtaining, in particular, that they are all nilpotent. Consequently, we provide classifications in low dimensions and identify the double extensions that maintain symplectic structures. By means of both elementary odd double extensions and generalized double extensions of quadratic symplectic Lie superalgebras, we obtain an inductive description of quadratic symplectic Lie superalgebras of filiform type.
Paper Structure (10 sections, 21 theorems, 94 equations)

This paper contains 10 sections, 21 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Basic concepts and previous results
  3. Basic concepts and some results
  4. Double extension of quadratic Lie superalgebras
  5. Symplectic extensions of nilpotent Lie superalgebras of filiform type
  6. $\delta_{\bar{1}}$-extension of symplectic Lie superalgebras
  7. Quadratic Symplectic Lie superalgebras: General case.
  8. Double extensions of quadratic symplectic Lie superalgebras
  9. Generalized double extensions of quadratic symplectic Lie superalgebras
  10. Inductive description of quadratic symplectic Lie superalgebras of filiform type

Key Result

Lemma 2.1

quadraticFLSA If ${\mathfrak g} = {\mathfrak g}_{\bar{0}} \oplus {\mathfrak g}_{\bar{1}}$ is a solvable Lie superalgebra with super-nilindex$(-,\dim({\mathfrak g}_{\bar{1}})),$ then ${\mathfrak g}_{\bar{1}}$ has filiform ${\mathfrak g}_{\bar{0}}$-module structure.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Remark 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 41 more