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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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[1] A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Math. Surveys and Monogr. Vol. 122 (Amer. Math. Soc., Providence, RI, 2005). · Zbl 1105.16001
[2] Frolova, Yu. Yu.; Shulezhko, O. V., Almost nilpotent varieties of Leibniz algebras, Prikl. Diskr. Matem., 2, 30, (2015)
[3] Mishchenko, S.; Valenti, A., An almost nilpotent variety of exponent 2, Isr. J. Math., 199, 241, (2014) · Zbl 1322.17001
[4] Shulezhko, O. V., New properties of almost nilpotent variety of exponent 2, Izv. Saratov Univ. (N.S.), Ser. Matem. Mekhan. Inform., 14, 316, (2014) · Zbl 1301.17003
[5] Mishchenko, S. P.; Shulezhko, O. V., Almost nilpotent varieties of arbitrary integer exponent, Vestnik Mosk. Univ., Matem. Mekhan., 2, 53, (2015) · Zbl 1394.17008
[6] Mishchenko, S. P.; Shulezhko, O. V., On almost nilpotent varieties in the class of commutative metabelian algebras, Vestnik Samar. Gos. Univ., Estest. Nauch. Ser., 3, 21, (2015) · Zbl 1394.17008
[7] Shulezhko, O. V., Almost nilpotent varieties in different classes of linear algebras, Chebyshevskii Sborn., 16, 67, (2015) · Zbl 1394.17008
[8] Zaicev, M. V.; Mishchenko, S. P., On the colength of varieties of linear algebras, Matem. Zametki, 79, 553, (2006) · Zbl 1114.17001
[9] Giambruno, A.; Mishchenko, S.; Zaicev, M., Codimensions of algebras and growth functions, Adv. Math., 217, 1027, (2008) · Zbl 1133.17001
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