Hall algebras associated to triangulated categories, II: almost associativity

TL;DR

This paper introduces an 'almost' associative multiplication for Hall algebras in 2-periodic triangulated categories and offers a new proof connecting these structures to Kac-Moody and elliptic Lie algebras.

Contribution

It establishes an 'almost' associative structure for Hall algebras in 2-periodic triangulated categories and provides a novel proof of a theorem relating these categories to Lie algebras.

Findings

Defined an 'almost' associative multiplication for indecomposable objects

Connected Hall algebra structures to symmetrizable Kac-Moody and elliptic Lie algebras

Provided a new proof of Peng and Xiao's theorem

Abstract

By using the approach in \cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost' associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in \cite{PX2000} which provides a way of realizing symmetrizable Kac-Moody algebras and elliptic Lie algebras via 2-periodic triangulated categories.