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On solvable Lie superalgebras of maximal rank

Bakhrom A. Omirov, Isamiddin S. Rakhimov, Gulkhayo O. Solijanova

Abstract

In this paper we establish some basic properties of superderivations of Lie superalgebras. Under certain conditions, for solvable Lie superalgebras with given nilradicals, we give estimates for upper bounds to dimensions of complementary subspaces to the nilradicals. Moreover, under these conditions we describe the solvable Lie superalgebras of maximal rank. Namely, we prove that an arbitrary solvable Lie superalgebra of maximal rank is isomorphic to the maximal solvable extension of nilradical of maximal rank.

On solvable Lie superalgebras of maximal rank

Abstract

In this paper we establish some basic properties of superderivations of Lie superalgebras. Under certain conditions, for solvable Lie superalgebras with given nilradicals, we give estimates for upper bounds to dimensions of complementary subspaces to the nilradicals. Moreover, under these conditions we describe the solvable Lie superalgebras of maximal rank. Namely, we prove that an arbitrary solvable Lie superalgebra of maximal rank is isomorphic to the maximal solvable extension of nilradical of maximal rank.
Paper Structure (5 sections, 22 theorems, 84 equations)

This paper contains 5 sections, 22 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some properties of superderivations
  4. Description of complex solvable Lie superalgebras of maximal rank
  5. Some maximal solvable extensions of nilpotent Lie superalgebras

Key Result

Theorem 2.3

Gilg The Lie superalgebra $\mathcal{L}=\mathcal{L}_0\oplus \mathcal{L}_1$ is nilpotent if and only if there exist positive integers $p$ and $q$ such that ${\mathcal{C}}^p(\mathcal{L}_0)={\mathcal{C}}^q(\mathcal{L}_1)=\{0\}.$

Theorems & Definitions (50)

  • Definition 2.1: Kac
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4: Engel's theorem
  • Example 2.5
  • Theorem 2.6
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • Theorem 3.4
  • ...and 40 more