PI-algebras with slow codimension growth

Let cn(A), n=1,2,… , be the sequence of codimensions of an algebra A over a field F of characteristic zero. We classify the algebras A (up to PI-equivalence) in case this sequence is bounded by a linear function. We also show that this property is closely related to the following: if ln(A), n=1,2,… , denotes the sequence of colengths of A, counting the number of Sn-irreducibles appearing in the nth cocharacter of A, then limn→∞ln(A) exists and is bounded by 2.