The Ringel--Hall Lie algebra of a spherical object

TL;DR

This paper investigates the algebraic structure of categories generated by spherical objects, determines their Picard groups, and characterizes associated Ringel--Hall Lie algebras, revealing their triangulated and orbit category properties.

Contribution

It explicitly computes the Picard group of categories generated by spherical objects and characterizes the Ringel--Hall Lie algebra for the 2-periodic case.

Findings

Picard group of $s_w$ determined

Orbit categories are triangulated and equivalent to derived categories of tubes

Ringel--Hall Lie algebra characterized for the 2-periodic case

Abstract

For an integer $w$, let $\cs_w$ be the algebraic triangulated category generated by a $w$-spherical object. We determine the Picard group of $\cs_w$ and show that each orbit category of $\cs_w$ is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. When $n=2$, the orbit category $\cs_w/\Sigma^2$ is 2-periodic triangulated, and we characterize the associated Ringel--Hall Lie algebra in the sense of Peng and Xiao.