Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Three-dimensional Lie algebras admitting regular semisimple algebraic Nijenhuis operators

Zhikhareva Ekaterina Sergeevna

Abstract

The aim of this paper is to classify all real and complex 3-dimensional Lie algebras admitting regular semisimple algebraic Nijenhuis operators. This problem is completely solved (see Theorems 2 and 3) by describing all Nijenhuis eigenbases for each 3-dimensional Lie algebra. It turns out that the answer is different in real and complex cases in the sence that there are real Lie algebras such that they do not admit an algebraic Nijenhuis operator, but their complexification admits such operators. An equally interesting question is to describe all algebraic Nijenhuis operators which are not equivalent by an automorphism of the Lie algebra. We give an answer to this question for some Lie algebras.

Three-dimensional Lie algebras admitting regular semisimple algebraic Nijenhuis operators

Abstract

The aim of this paper is to classify all real and complex 3-dimensional Lie algebras admitting regular semisimple algebraic Nijenhuis operators. This problem is completely solved (see Theorems 2 and 3) by describing all Nijenhuis eigenbases for each 3-dimensional Lie algebra. It turns out that the answer is different in real and complex cases in the sence that there are real Lie algebras such that they do not admit an algebraic Nijenhuis operator, but their complexification admits such operators. An equally interesting question is to describe all algebraic Nijenhuis operators which are not equivalent by an automorphism of the Lie algebra. We give an answer to this question for some Lie algebras.

Paper Structure

This paper contains 6 sections, 3 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Proof of Theorem \ref{['t1']}
  4. Proof of Theorem \ref{['t2']}
  5. Proof of Theorem \ref{['t3']}
  6. Remark on equivalence of algebraic Nijenhuis operators

Key Result

Theorem 1

Real or complex Lie algebra $\mathfrak g$ admits a regular semisimple algebraic Nijenhuis operator if and only if there exists a basis $\eta_1, \dots, \eta_n$ such that In other words, this condition means that any two basic vectors $\eta_i, \eta_j$ generate a two-dimensional Lie subalgebra in $\mathfrak g$.

Theorems & Definitions (6)

  • Example 1
  • Example 2
  • Theorem 1
  • Example 3
  • Theorem 2
  • Theorem 3