Classification of complete left-invariant affine structures on the Oscillator group

TL;DR

This paper introduces a method based on left-symmetric algebra extensions to classify complete left-invariant affine structures on the oscillator group, a specific solvable Lie group, enhancing understanding of affine geometries on such groups.

Contribution

It develops a novel classification method using algebra extension theory and applies it specifically to the oscillator group, providing new insights into affine structures on solvable Lie groups.

Findings

Classified all complete left-invariant affine structures on the oscillator group

Demonstrated the effectiveness of the extension-based classification method

Enhanced understanding of affine geometries on solvable Lie groups

Abstract

The goal of this paper is to provide a method, based on the theory of extensions of left-symmetric algebras, for classifying left-invariant affine structures on a given solvable Lie group of low dimension. To better illustrate our method, we shall apply it to classify complete left-invariant affine structures on the oscillator group.