A characterization of varieties of associative algebras of exponent two (Q2731891)

Recently the authors of the paper under review [\textit{A. Giambruno, M. Zaicev}, Adv. Math. 140, No. 2, 145-155 (1998; Zbl 0920.16012); ibid. 142, No. 2, 221-243 (1999; Zbl 0920.16013)] have established the remarkable result that the sequence of the codimensions of the polynomial identities \(c_n(R)\), \(n=1,2,\ldots\), of any associative PI-algebra \(R\) over a field of characteristic 0 has integral exponential growth, i.e. \(\exp(R)=\lim_{n\to\infty}(c_n(R))^{1/n}\) exists and is an integer.NEWLINENEWLINENEWLINEThe algebras of exponent 0 are nilpotent. Those of exponent 1 (i.e. with polynomial growth of the codimensions) have been described by \textit{A. R. Kemer} [Sib. Mat. Zh. 19, 54-69 (1978; Zbl 0385.16009), English translation: Sib. Math. J. 19, 37-48 (1978); and Varieties of finite rank (Russian), in Proc. 15-th All-Union algebra conf., Krasnoyarsk, vol. 2 (1979), p. 73]. In particular, \(\exp(R)=1\) if and only if \(R\) satisfies polynomial identities \(f=0\) and \(g=0\) which do not hold for the Grassmann algebra and the algebra of \(2\times 2\) upper triangular matrices, respectively. In the present paper the authors go further and describe all PI-algebras such that \(\exp(R)=2\). They give a list of 5 algebras \(A_1,\ldots,A_5\) (all appearing naturally in the PI-theory) such that \(\exp(R)>2\) if and only if some \(A_i\) satisfies all polynomial identities of \(R\).