Poincare´-Hilbert Series and Gel’fand-Kirillov Dimension
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 Hilbert series of an algebra, otherwise called the Poincare´-Hilbert series, and the important invariant known as the Gel’fand-Kirillov dimension, abbreviated as GK dim. The GK dim of an affine PI-algebra A always exists (and is bounded by the Shirshov height), and is an integer when A is representable. (However, there are examples of affine PI-algebras with non-integral GK dim.) These reasonably decisive results motivate us to return to the Hilbert 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.