Ideals in the Goldman Algebra

TL;DR

This paper investigates the ideals of the Goldman Lie algebra by constructing a homomorphism to a simpler algebraic structure, classifying ideals in different module contexts, and revealing an infinite chain of unique ideals.

Contribution

It introduces a novel approach to classify and understand the ideals of the Goldman Lie algebra through algebra homomorphisms and module analysis.

Findings

Existence of an infinite class of ideals containing specific free homotopy classes in the integer module case.

Complete classification of ideals in the rational module case.

Identification of an infinite chain of ideals not derived from the simpler structure.

Abstract

The goal of this work is to study the ideals of the Goldman Lie algebra $S$. To do so, we construct an algebra homomorphism from $S$ to a simpler algebraic structure, and focus on finding ideals of this new structure instead. The structure $S$ can be regarded as either a $\mathbb{Q}$-module or a $\mathbb{Q}$-module generated by free homotopy classes. For $\mathbb{Z}$-module case, we proved that there is an infinite class of ideals of $S$ that contain a certain finite set of free homotopy classes. For $\mathbb{Q}$-module case, we can classify all the ideals of the new structure and consequently obtain a new class of ideals of the original structure. Finally, we show an interesting infinite chain of ideals that are not those ideals obtained by considering the new structure.