Almost nilpotent varieties with non-integer exponents do exist. (English. Russian original) Zbl 1370.17004

Mosc. Univ. Math. Bull. 71, No. 3, 115-118 (2016); translation from Vestn. Mosk. Univ., Ser. I 71, No. 3, 42-46 (2016).
For a variety \(\mathbf{ V}\) of (nonassociative) algebras over a field of characteristic 0 the lower and upper exponent of \(\mathbf{ V}\) is defined, respectively, by \(\underline{\exp}(\mathbf{ V})=\liminf_{n\to\infty}\root n\of{c_n(\mathbf{ V})}\) and \(\overline{\exp}(\mathbf{ V})=\limsup_{n\to\infty}\root n\of{c_n(\mathbf{ V})}\), where \(c_n(\mathbf{ V})\) is the \(n\)-th codimension of \(\mathbf{ V}\). In the paper under review the author constructs a three-generated nonassociative algebra \(A\) with the following property. For any nonnilpotent subvariety \(\mathbf{ U}\) of the variety \(\text{var}(A)\) generated by \(A\) the upper and lower exponents satisfy the inequality \(\Phi(1/6)\leq\underline{\exp}(\mathbf{ U})\leq\overline{\exp}(\mathbf{ U}) \leq\Phi(1/3)\), where \(\Phi(\alpha)=1/\alpha^{\alpha}(1-\alpha)^{1-\alpha}\), \(0<\alpha<1/2\). Since \(1<\Phi(1/6)<\Phi(1/3)<2\), this immediately implies that \(\text{var}(A)\) has a nonnilpotent subvariety with nonintegral exponents and with the property that all its proper subvarieties are nilpotent.


17A30 Nonassociative algebras satisfying other identities
17A50 Free nonassociative algebras
16P90 Growth rate, Gelfand-Kirillov dimension
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