Analytic isomorphisms of compressed local algebras

In this paper we consider Artin compressed local algebras, that is local algebras with maximal length in the class of those with given embedding dimension and socle type. They have been widely studied by several authors, among others by Boij, Iarrobino, Fr\"{o}berg and Laksov. In this class the Gorenstein algebras play an important role. The authors proved that a compressed Gorenstein $K$-algebra of socle degree $3 $ is canonically graded, i.e. analytically isomorphic to its associated graded ring, see \cite{ER12}. This unexpected result has been extended to compressed level $K$-algebras of socle degree $3$ in \cite{DeS12}. This paper somehow concludes the investigation proving that Artin compressed Gorenstein $K$-algebras of socle degree $s \le 4$ are always canonically graded, but explicit examples prove that the result does not extend to socle degree $5 $ or to compressed level $K$-algebras of socle degree $4$ and type $>1.$ As a consequence of this approach we present classes of Artin compressed $K$-algebras which are canonically graded.