Eventually non-decreasing codimensions of *-identities
Let \(A\) be an algebra (not necessarily associative) over a field \(F\) of characteristic 0. The polynomial identities of \(A\) are completely determined by the multilinear ones. Let \(F\langle X\rangle\) be the free associative algebra over \(F\) that is freely generated by the set of variables \(X=\{x_1,x_2, \ldots\}\), and denote by \(P_n\) its vector subspace of all multilinear polynomials in \(x_1\), \dots, \(x_n\). If \(\operatorname{Id}(A)\) is the T-ideal of \(A\) then one can study \(P_n\cap \operatorname{Id}(A)\), \(n\ge 1\), in order to know \(\operatorname{Id}(A)\). As \(P_n\) is a left module over the symmetric group \(S_n\), and \(\operatorname{Id}(A)\) is closed under permutations of the variables, one has that \(P_n\cap \operatorname{Id}(A)\) is a submodule. Hence the representation theory of \(S_n\) can be employed in the study of PI algebras. It turns out it is much more convenient to study the factor module \(P_n/(P_n\cap \operatorname{Id}(A))\). Its dimension \(c_n(A)\) is the \(n\)th codimension of \(A\) (of \(\operatorname{Id}(A)\), more precisely). The codimension sequence of a PI algebra was introduced and first studied by \textit{A. Regev} [Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)]. Regev proved that if \(A\) satisfies an identity of degree \(d\) then \(c_n(A)\le (d-1)^{2n}\) for every \(n\), and applied this fact in order to prove that the tensor product of PI algebras is PI as well. The fact that the growth of \(c_n(A)\) is bounded by an exponential function implies that \(P_n\cap \operatorname{Id}(A)\) becomes very large if \(\operatorname{Id}(A)\ne 0\). Furthermore it was proved by \textit{A. Giambruno} and \textit{M. Zaicev} [Adv. Math. 140, No. 2, 145--155 (1998; Zbl 0920.16012); Adv. Math. 142, No. 2, 221--243 (1999; Zbl 0920.16013)], that \(\lim_{n\to\infty} (c_n(A)^{1/n})\) always exists and is a nonnegative integer, called the PI exponent of \(A\), \(\exp(A)\). It is an important problem to determine more precisely the growth of the codimension sequence of a PI algebra. \textit{A. Berele} [Adv. Appl. Math. 41, No. 1, 52--75 (2008; Zbl 1145.05052)] proved that for every PI algebra \(A\) there exist positive constants \(C_1\) and \(C_2\) such that \(C_1 n^t q^n \le c_n(A) \le C_2 n^t q^n\) as long as the codimension sequence of \(A\) is eventually non-decreasing. Later on \textit{A. Giambruno} and \textit{M. Zaicev} [Bull. Lond. Math. Soc. 46, No. 4, 771--778 (2014; Zbl 1302.16015)] proved that this is the case for every associative algebra. Recall here that \(q\) is the PI exponent of \(A\), and \(t=\lim_{n\to\infty} \log_n (c_n(A)/q^n)\) is an integer or half integer. The latter paper by Giambruno and Zaicev contains an example of a nonassociative algebra whose codimensions do not form an eventually non-decreasing sequence. Recently \textit{A. Giambruno} and \textit{D. La Mattina} [Isr. J. Math. 239, No. 1, 1--20 (2020; Zbl 1459.16024)] established that in the case of associative algebras with involution, the codimension sequence is eventually non-decreasing. The paper under review studies a similar problem in the nonassociative setting. The authors construct an algebra with involution \(B\) whose codimension sequence is not eventually non-decreasing.
scientific article; zbMATH DE number 7348117