ABSTRACT
Our aim in this chapter is to study growth of Pi-algebras by means of the techniques developed so far. To do this, we shall define the Hubert series of an algebra, otherwise called the Poincaré-Hilbert series, and the important invariant known as the Gelfand-Kirillov dimension. The Gelfand-Kirillov dimension of an affine Pi-algebra A always exists (and is bounded by the Shirshov height), and is an integer when A is represent able. (However, there are examples of affine PL-algebras with non-integral Gelfand-Kirillov dimension.) These reasonably decisive results motivate us to return to the Hubert series, especially in determining when it is a rational function. We shall discuss (in brief) some important ties of the Hilbert series to the codimension sequence.
