Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let k be a fixed algebraically closed field of arbitrary characteristic,let Λ be a finite dimensional self-injective k-algebra,and let V be an indecomposable non-projective left Λ-module with finite dimension over k.We prove that if τΛV is the Auslander-Reiten translation of V,then the versal deformation rings R(Λ,V)and R(Λ,τΛV)(in the sense of F.M.Bleher and the second author)are isomorphic.We use this to prove that if Λ is further a cluster-tilted k-algebra,then R(Λ,V)is universal and isomorphic to k.