The author estimates the Gel’fand-Kirillov dimension \(\text{GK}\dim(A)\) of the relatively free associative algebra \(A\) over an arbitrary ground field. This dimension is determined by the complexity type of the algebra \(A\) or by a set of semidirect products of matrix algebras over the ring of polynomials from the variety \(\text{Var}(A)\).
The following main theorem is established: For an \(s\)-generated relatively free algebra \(A\), \(\text{GK}\dim(A)\) depends only on its complexity type, that is, \(\text{GK}\dim(A)\) is equal to the maximum Gel’fand-Kirillov dimension of a semidirect product of the algebras of generic matrices from \(\text{Var}(A)\). The Gel’fand-Kirillov dimension of such a semidirect product is equal to the sum of the Gel’fand-Kirillov dimensions of its factors. The complexity type of the algebra \(A\) is the same as that of the subalgebra generated by two of its generators.
The result is proven in several steps. First, the author studies the semidirect products and their applications to the calculation of the Gel’fand-Kirillov dimension of the relatively free representable algebras. Then the relatively free associative case is reduced to the representable case, using the technique of representable spaces, that are extensions of a representable algebra by nilpotents. The semidirect products of matrix algebras are then addressed.