Superalgebras and algebras with involution: classifying varieties of quadratic growth

Let F be a field of characteristic zero. By a phi-algebra we mean a superalgebra or an algebra with involution over F: In the last years, the sequence of phi-codimensions {c(n)(phi)(A)}(n >= 1) of a phi-algebra A has been extensively studied. In this paper, we classify varieties generated by unitary phi-algebras having quadratic growth of phi-codimensions. As a consequence we obtain that a unitary phi-algebra with quadratic growth is T-phi-equivalent to a finite direct sum of minimal unitary phi-algebras with at most quadratic growth of the u-codimensions. In addition, we explicit all quadratic functions describing the u-codimension sequence of a unitary phi-algebra.