On root categories of finite-dimenisonal algebras

TL;DR

This paper explores the structure of root categories of finite-dimensional algebras with finite global dimension, linking them to Ringel-Hall Lie algebras and providing new insights into GIM Lie algebras.

Contribution

It introduces a new perspective on root categories via triangulated hulls and applies this to study Ringel-Hall Lie algebras for specific algebras, addressing questions on GIM Lie algebras.

Findings

Root categories are characterized as triangulated hulls of 2-periodic orbit categories.

Ringel-Hall Lie algebras for certain algebras with global dimension 2 are constructed.

Provides an alternative answer to a question on GIM Lie algebras by Slodowy.

Abstract

For any finite-dimensional algebra $A$ over a field $k$ with finite global dimension, we investigate the root category $\cR_A$ as the triangulated hull of the 2-periodic orbit category of $A$ via the construction of B. Keller in "On triangulated orbit categories". This is motivated by Ringel-Hall Lie algebras associated to 2-periodic triangulated categories. As an application, we study the Ringel-Hall Lie algebras for a class of finite-dimensional $k$-algebras with global dimension 2, which turn out to give an alternative answer for a question of GIM Lie algebras by Slodowy in "Beyond Kac-Moody algebra, and inside".