The Ringel--Hall Lie algebra of a spherical object

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.Comment: 26page

Exchange graphs of cluster algebras have the non-leaving-face property

The claim in the title is proved