The GK dimension of relatively free algebras of PI-algebras
Let \(\mathfrak{M}\) be a variety of \(PI\)-algebras, \(A=A_n\in\mathfrak{M}\) be an \(n\)-generated relatively free algebra in \(\mathfrak{M}\), \(V_A(k)\) is dimension of space generated by monomials of length \(n\). It is known that \(V_A(k)\sim k^{GK(A_n)}\), where \(GK(A)\) is the Gelfand-Kirillov dimension of an algebra \(A\). This dimension was calculated in the paper [\textit{A. Ya. Belov}, Sb. Math. 195, No. 12, 1703--1726 (2004; Zbl 1090.16007); translation from Mat. Sb. 195, No. 12, 3--26 (2004)]. The Hilbert series \(H_A(t)=\sum_n V_A(n)t^n\) are found to be rational in [\textit{A. Ya. Belov}, Russ. Math. Surv. 52, No. 2, 394--395 (1997; Zbl 0919.16018); translation from Usp. Mat. Nauk 52, No. 2, 153--154 (1997)]. For a graded version of this result see [\textit{E. Aljadeff} and the reviewer, Bull. Lond. Math. Soc. 44, No. 3, 520--532 (2012; Zbl 1259.16023)]. The notion of Hilbert series can be designed to the infinitely generated case for abstract varieties. Consider the relatively free algebra \(A_\infty \in\mathfrak{M}\) with countable number of generators \(x_1,\dots,x_n\dots\). Let \(c_n(\mathfrak{M})\) be the dimension of a space generated by multilinear words of length \(n\) over \(x_1,\dots,x_n\) The codimension series \(\sum c_n(\mathfrak{M}) t^n\) were introduced by A. Regev; it is well known that \(c_n(\mathfrak{M})\sim \lambda^n\), and \(\lambda=\mathrm{Exp}(\mathfrak{M})\) is called the \textit{exponent} of \(\mathfrak{M}\). There is a celebrated series of papers of Giambruno, Zaicev, Mishcenko calculating the exponent in terms of underlying algebras. The rationality result can be generalized somehow to codimension sequences [\textit{A. Berele}, J. Algebra 298, No. 1, 208--214 (2006; Zbl 1106.16024)]. The paper develops this deep relation and provides graded version of it. The author proves a strict relation between the Gelfand-Kirillov (GK) dimension of the relatively free (graded) algebra of a PI-algebra and its (graded) exponent. As a consequence, he shows a Bahturin-Zaicev type result relating the \(GK\) dimension of the relatively free algebra of a graded PI-algebra and the one of its neutral part. He also deduces that the growth of the relatively free graded algebra of a matrix algebra is maximal when the grading is fine. Finally, he computes the graded GK dimension of the matrix algebra with any grading and the graded GK dimension of any verbally prime algebra endowed with an elementary grading. The paper is elegant and nicely written.
scientific article; zbMATH DE number 7032793
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