Subcategories of Fixed Points of Mutations by Exceptional Objects in Triangulated Categories

We first prove that the subcategory of fixed points of mutation determined by an exceptional object E in a triangulated category coincide with the perpendicular category of E.Based on this characterisation,we prove that the subcategory of fixed points of mutation in the derived category of the coherent sheaves on weighted projective line with genus one is equivalent to the derived category of a hereditary algebra.Meanwhile,we induce two new recollements by left and right mutations from a given recollement.