On almost nilpotent varieties of subexponential growth. (English) Zbl 1333.17001

Summary: Let \({}_2\mathcal{N}\) be the variety of left-nilpotent algebras of index two, that is the variety of algebras satisfying the identity \(x(y z) \equiv 0\). We introduce two new varieties, denoted by \(\mathcal{V}_{\mathit{sym}}\) and \(\mathcal{V}_{\mathit{alt}}\), contained in the variety \({}_2\mathcal{N}\) and we prove that \(\mathcal{V}_{\mathit{sym}}\) and \(\mathcal{V}_{\mathit{alt}}\) are the only two varieties almost nilpotent of subexponential growth.


17A30 Nonassociative algebras satisfying other identities
Full Text: DOI


[1] Berele, A.; Regev, A., Applications of hook Young diagrams to P.I. algebras, J. Algebra, 82, 559-567, (1983) · Zbl 0517.16013
[2] Drensky, V., Free algebras and PI-algebras. graduate course in algebra, (2000), Springer Singapore · Zbl 0936.16001
[3] Drensky, V.; Piacentini Cattaneo, G. M., Varieties of metabelian Leibniz algebras, J. Algebra Appl., 1, 1, 31-50, (2002) · Zbl 1025.17001
[4] Giambruno, A.; Mishchenko, S. P., Irreducible characters of the symmetric group and exponential growth
[5] Giambruno A, A.; Mishchenko, S.; Zaicev, M., Algebras with intermediate growth of the codimensions, Adv. in Appl. Math., 37, 3, 360-377, (2006) · Zbl 1111.16022
[6] Giambruno, A.; Mishchenko, S.; Zaicev, M., Codimensions of algebras and growth functions, Adv. Math., 217, 1027-1052, (2008) · Zbl 1133.17001
[7] Giambruno, A.; Zaicev, M., On codimension growth of finitely generated associative algebras, Adv. Math., 140, 145-155, (1998) · Zbl 0920.16012
[8] Giambruno, A.; Zaicev, M., Exponential codimension growth of P.I. algebras: an exact estimate, Adv. Math., 142, 221-243, (1999) · Zbl 0920.16013
[9] Giambruno, A.; Zaicev, M., Polynomial identities and asymptotic methods, Math. Surveys Monogr., vol. 122, (2005), AMS Providence, RI · Zbl 1105.16001
[10] James, G.; Kerber, A., The representation theory of the symmetric group, Encyclopedia Math. Appl., vol. 16, (1981), Addison-Wesley London
[11] Mishchenko, S. P., Varieties of linear algebras with colength one, Moscow Univ. Math. Bull., 65, 23-27, (2010) · Zbl 1304.17002
[12] Mishchenko, S. S., New example of a variety of Lie algebras with fractional exponent, Vestnik Moskov. Univ. Ser. I Mat. Mekh., Moscow Univ. Math. Bull., 66, 6, 264-266, (2011), English translation in: · Zbl 1304.17016
[13] Mishchenko, S. P.; Valenti, A., Varieties with at most quadratic growth, Israel J. Math., 178, 209-228, (2010) · Zbl 1223.17007
[14] Mishchenko, S. P.; Valenti, A., An almost nilpotent variety of exponent 2, Israel J. Math., 199, 241-258, (2014) · Zbl 1322.17001
[15] Mishchenko, S. P.; Shulezhko, O. V., An almost nilpotent variety of any integer exponent, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (2014), (in Russian). Article in press
[16] Mishchenko, S.; Zaicev, M., An example of a variety of Lie algebras with a fractional exponent, Algebra, J. Math. Sci. (N. Y.), 93, 977-982, (1999) · Zbl 0933.17004
[17] Petrogradskii, V. M., Growth of polynilpotent varieties of Lie algebras, and rapidly increasing entire functions, Mat. Sb., Sb. Math., 188, 913-931, (1997), English translation in: · Zbl 0890.17002
[18] Regev, A., Existence of identities in \(A \otimes B\), Israel J. Math., 11, 131-152, (1972) · Zbl 0249.16007
[19] Zaicev, M. V., Integrality of exponents of growth of identities of finite-dimensional Lie algebras, Izv. Ross. Akad. Nauk Ser. Mat., Izv. Math., 66, 463-487, (2002), English translation in: · Zbl 1057.17003
[20] Zaicev, M. V., The existence of PI-exponents growth of codimension, (International Conference Mal’tsev Readings, 11-15 November 2013, Novosibirsk, (2013)), 130, (in Russian)
[21] Zaicev, M. V.; Mishchenko, S. P., An example of a variety of linear algebras with fractional polynomial growth, Vestnik Moskov. Univ. Ser. I Mat. Mekh., Moscow Univ. Math. Bull., 63, 1, 27-32, (2008), 71 (in Russian); Translation in:
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.