Maximal PI-exponents of finite-dimensional algebras (Q6076551)

If \(F\) is a field of characteristic \(0\) and \(A\) is a (not necessarily associative) \(F\)-algebra then all polynomial identities of \(A\) follow from the multilinear ones. One of the main ways to measure how many are the polynomial identities of \(A\) is to study the sequence \(c_n(A)\), \(n=1,2,\ldots\), of the (multilinear) codimensions. In the case of associative PI-algebras the already classical result of \textit{A. Regev} [Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)] gives that the codimension sequence is exponentially bound. \textit{A. Giambruno} and \textit{M. Zaicev} [Adv. Math. 140, No. 2, 145--155 (1998; Zbl 0920.16012); ibid. 142, No. 2, 221--243 (1999; Zbl 0920.16013)] confirmed a conjecture of Amitsur that the PI-exponent \(\exp(A)=\lim_{n\to\infty}\sqrt[n]{c_n(A)}\) of \(A\) exists and is a nonnegative integer. The picture is different if one considers Lie algebras. \textit{I. B. Volichenko} [Sib. Math. J. 25, 370--382 (1984; Zbl 0575.17006)] showed that the variety \({\mathfrak A}{\mathfrak N}_2\) defined by the identity \([[x_1,x_2,x_3],[x_4,x_5,x_6]]=0\) is of exponential growth and \textit{M. V. Zaĭtsev} and \textit{S. P. Mishchenko} [J. Math. Sci., New York 93, No. 6, 977--982 (1999; Zbl 0933.17004)] constructed an example of a Lie algebra with a fractional PI-exponent. On the other hand, \textit{M. V. Zaitsev} [Izv. Math. 66, No. 3, 463--487 (2002; Zbl 1057.17003)] proved that \(\exp(A)\) is a nonnegative integer for any finite dimensional Lie algebra \(A\). Both in the case of finite dimensional associative and Lie algebras the condition \(\exp(A)=\dim(A)\) is equivalent to the simplicity of \(A\). (By a result of Bahturin and the reviewer [\textit{Y. Bahturin} and \textit{V. Drensky}, Linear Algebra Appl. 357, No. 1--3, 15--34 (2002; Zbl 1019.16011)] \(\limsup_{n\to\infty}\sqrt[n]{c_n(A)}\) is bounded from above by \(\dim(A)\) and hence if \(\exp(A)\) exists then \(\exp(A)\leq\dim(A)\).) The main result of the paper under review is the construction of a nonassociative algebra \(A_m\) of dimension \(m+1\), \(m\geq 3\), which is not simple and \(\exp(A_m)=\dim(A_m)=m+1\).