On flat deformations and their applications

We say that a formal deformation from an algebra N to algebra A is strongly flat if for every real number e there is a real number $0<s<e$ such that this deformation specialised at $t=s$ gives an algebra isomorphic to A. We show that every strongly flat deformation from a finite-dimensional C-algebra N to a semisimple C-algebra A specialised at $t=s$ for all sufficiently small real numbers $s>0$ gives an algebra isomorphic to A. We also give a characterisation of semisimple algebras A to which a given algebra N cannot be deformed to. This gives a partial answer to a question of Michael Wemyss on Acons [14]. We also give a partial answer to question 6.5 from [1].