Three-dimensional Lie algebras admitting regular semisimple algebraic Nijenhuis operators

TL;DR

This paper classifies all 3-dimensional real and complex Lie algebras that admit regular semisimple algebraic Nijenhuis operators, providing a complete description of their eigenbases and differences between real and complex cases.

Contribution

It provides a complete classification of 3D Lie algebras with algebraic Nijenhuis operators, including eigenbasis descriptions and distinctions between real and complex cases.

Findings

Some real Lie algebras do not admit Nijenhuis operators.

Complexification of certain real Lie algebras admits Nijenhuis operators.

Complete description of Nijenhuis eigenbases for all 3D Lie algebras.

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.