Varieties with at most quadratic growth (Q607832)

The authors study varieties of non-necessarily associative algebras with polynomial growth of the codimension sequence, over a field of characteristic zero. It is well known that for associative and Lie algebras the varieties with polynomial growth of codimensions behave very nicely, which does not hold for varieties of arbitrary linear algebras. Recently \textit{M. V. Zaĭtsev} and \textit{S. P. Mishchenko} [Mosc. Univ. Math. Bull. 63, No. 1, 27--32 (2008); translation from Vestn. Mosk. Univ., Ser. I 2008, No. 1, 25--31 (2008; Zbl 1199.17001)] constructed for any real \(\alpha\), \(3<\alpha<4\), a variety \(\mathcal V\) such that its codimensions satisfy \(C_1<n^{\alpha}<c_n({\mathcal V})<C_2n^{\alpha}\) for suitable positive constants \(C_1\) and \(C_2\). In the present paper the authors study varieties \(\mathcal V\) with codimensions satisfying \(c_n({\mathcal V})<Cn^{\alpha}\), \(0<\alpha<2\). They show that if \(\alpha<1\), then \(c_n({\mathcal V})\leq 1\) for \(n\) large enough. If \(1<\alpha<2\) and \(\mathcal V\) satisfies the commutative law (but still \(\mathcal V\) is non-associative), then either \(c_n({\mathcal V})\leq 1\) for \(n\) large enough or \(\lim_{n\to\infty} \log_n(c_n({\mathcal V}))= 1\).