ON ALMOST POLYNOMIAL GROWTH OF PROPER CENTRAL POLYNOMIALS

To any associative algebra $A$ is associated a numerical sequence $c_n^{\delta}(A)$, $n\ge 1$, called the sequence of proper central codimensions of $A$. It gives information on the growth of the proper central polynomials of the algebra. If $A$ is a PI-algebra over a field of characteristic zero it has been recently shown that such a sequence either grows exponentially or is polynomially bounded. Here we classify, up to PI-equivalence, the algebras $A$ for which the sequence $c_n^{\delta}(A)$, $n\ge 1$, has almost polynomial growth. We prove that the prove that the Grassmann algebra and two special algebras of upper triangular matrices Then we face a similar problem in the setting of group-graded algebras and we obtain a classification also in this case when the corresponding sequence of proper central codimensions has almost polynomial growth.