# Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators

## TL;DR

This paper investigates the structure of Nijenhuis operators at scalar-type points, linking their tangent spaces to left-symmetric algebras, and classifies non-degenerate cases especially in two dimensions, advancing understanding of their linearization.

## Contribution

It establishes a connection between Nijenhuis operators at scalar points and left-symmetric algebras, providing classifications in low dimensions and addressing the linearization problem.

## Key findings

- Classification of non-degenerate left-symmetric algebras in 2D for smooth and analytic categories
- Complete classification of 2D real left-symmetric algebras
- Identification of differences between smooth and analytic cases

## Abstract

A field of endomorphisms $R$ is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of $R$ called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or Vinberg-Kozul algebras). Following Weinstein's approach to linearization of Poisson structures, we state the linearisation problem for Nijenhuis operators and give an answer in terms of non-degenerate left-symmetric algebras. In particular, in dimension 2, we give classification of non-degenerate left-symmetric algebras for the smooth category and, with some small gaps, for the analytic one. These two cases, analytic and smooth, differ. We also obtain a complete classification of two-dimensional real left-symmetric algebras, which may be an interesting result on its own. This work is the second part of a series of papers on Nijenhuis Geometry started with arXiv:1903.04603