## 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.

### MSC:

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

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