Sparseness of t-structures and negative Calabi-Yau dimension in triangulated categories generated by a spherical object

Let k be an algebraically closed field and let T be the k-linear algebraic triangulated category generated by a w-spherical object for an integer w. For certain values of w this category is classical. For instance, if w = 0 then it is the compact derived category of the dual numbers over k. As main results of the paper we show that for w \leq 0, the category T has no non-trivial t-structures, but does have one family of non-trivial co-t-structures, whereas for w \geq 1 the opposite statement holds. Moreover, without any claim to originality, we observe that for w \leq -1, the category T is a candidate to have negative Calabi-Yau dimension since \Sigma^w is the unique power of the suspension functor which is a Serre functor.