Varieties of unitary algebras with small growth of codimensions

In the last years, the sequence of codimensions of PI-algebras has been studied by several authors and the classification of unitary algebras, up to equivalence, with at most cubic codimension growth was given by Giambruno, La Mattina and Petrogradsky in 2007. In this paper, we establish a new approach by studying the possibilities of specific proper codimensions of a unitary algebra with growth n4 in order to present a complete list of varieties generated by unitary algebras with polynomial growth n4. Also, we classify, up to PI-equivalence, the unitary algebras with growth n5 whose leading coefficient of the polynomial describing the codimension sequence achieves the largest and the smallest possible value.