  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.
