On codimension growth of finitely generated associative algebras
Let \(A\) be a PI-algebra over a field \(F\) of characteristic 0 and let \(c_n(A)\) be the \(n\)-th codimension of (the multilinear polynomial identities of) \(A\), \(n=1,2,\ldots\). Starting in the early 70's when Regev used his exponential bounds of the codimensions to show that the tensor product of two PI-algebras is PI again, a lot of investigations have been devoted to the asymptotic behaviour of the codimension sequences both in the case of arbitrary PI-algebras and in the case of some special PI-algebras important from a different point of view. As a first approximation to the more precise estimates of the growth of the codimensions of a PI-algebra \(A\) one may study the behaviour of \(\root n\of{c_n(A)}\). A well known conjecture among the mathematicians working in PI-theory is that \(\text{Inv}(A)=\lim_{n\to\infty}\root n\of{c_n(A)}\) always exists and is an integer. In a series of two important papers [the second being \textit{A. Giambruno, M. Zaicev}, Adv. Math. 142, No. 2, 221-243 (1999; reviewed below, Zbl 0920.16013)], the authors give the affirmative answer to the above conjecture. In the paper under review they handle the case of finitely generated PI-algebras. Since the theory of Kemer gives that the polynomial identities of any finitely generated PI-algebra coincide with those of some finite dimensional algebra, it is sufficient to consider the case when \(\dim_FA<\infty\). The authors describe an effective procedure based on the Wedderburn-Malcev decomposition (\(A=B+J\), where \(J\) is the Jacobson radical of \(A\) and \(B=B_1\oplus\cdots\oplus B_n\) is the decomposition of the semisimple part of \(A\) into simple components) how to find explicitly \(\text{Inv}(A)\). Several interesting consequences of the main result are obtained as well. In particular, the finite dimensional algebra \(A\) is central simple over the base field \(F\) if and only if \(\text{Inv}(A)=\dim_FA\).
scientific article; zbMATH DE number 1268547