On Riedtmann's Lie algebra of the gentle one-cycle algebra $\Lambda(n-1,1,1)$

TL;DR

This paper computes an extended version of Riedtmann's Lie algebra for a specific gentle one-cycle algebra and demonstrates its decomposition according to the positive roots of a type BC root system.

Contribution

It extends Riedtmann's Lie algebra for the algebra $ ext{Lambda}(n-1,1,1)$ and reveals its Cartan decomposition related to type BC root systems.

Findings

Extended Riedtmann's Lie algebra computed

Admits a Cartan decomposition by positive roots of type BC

Connects algebra structure with root system theory

Abstract

An extended version of Riedtmann's Lie algebra of the gentle one-cycle algebra $\Lambda(n-1,1,1)$ is computed and is shown to admit a Cartan decomposition by the positive roots of the root system of type $\mathbf{BC}_n$.