×

zbMATH — the first resource for mathematics

An uncountable family of almost nilpotent varieties of polynomial growth. (English) Zbl 1431.16019
Summary: A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of
1) a countable family of almost nilpotent varieties of at most linear growth and
2) an uncountable family of almost nilpotent varieties of at most quadratic growth.

MSC:
16P90 Growth rate, Gelfand-Kirillov dimension
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
17A50 Free nonassociative algebras
20C30 Representations of finite symmetric groups
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Chang, N. T.K.; Frolova, Yu. Yu., Commutative metabelian almost nilpotent varieties growth not higher than exponential, (International Conference Mal’tsev Readings: 10-13 november 2014, Novosibirsk, (2014)), 119, (in Russian). Available at:
[2] Drensky, V., Free algebras and PI-algebras. graduate course in algebra, (2000), Springer Singapore · Zbl 0936.16001
[3] Frolova, Yu. Yu.; Shulezhko, O. V., Almost nilpotent varieties of Leibniz algebras, Priklad. Discret. Mat., 2, 28, 30-36, (2015), (in Russian)
[4] Giambruno, A.; Mishchenko, S., Polynomial growth of the codimensions: a characterization, Proc. Am. Math. Soc., 138, 3, 853-859, (March 2010)
[5] Giambruno, A.; Mishchenko, S.; Zaicev, M., Algebras with intermediate growth of the codimensions, Adv. Appl. Math., 37, 360-377, (2006) · Zbl 1111.16022
[6] Giambruno, A.; Zaicev, M., Polynomial identities and asymptotic methods, Math. Surv. Monogr., vol. 122, (2005), AMS Providence, R.I. · Zbl 1105.16001
[7] Lothaire, M., Algebraic combinatorics on words, Encycl. Math. Appl., vol. 90, (2002), Cambridge University Press Cambridge · Zbl 1001.68093
[8] Mishchenko, S. P., Almost nilpotent varieties with non-integer exponents do exist, Vestn. Mosk. Univ., Ser. Filos., 71, 3, 42-46, (2016), (in Russian); translation in Moscow University Mathematics Bulletin 71 (3), 115-118 · Zbl 1370.17004
[9] Mishchenko, S. P.; Shulezhko, O. V., Description almost nilpotent anticommutative metabelian varieties with subexponential growth, (International Conference Mal’tsev Readings: 10-13 november 2014, Novosibirsk, (2014)), (in Russian). Available at:
[10] Mishchenko, S. P.; Shulezhko, O. V., Almost nilpotent varieties of arbitrary integer exponent, Vestn. Mosk. Univ., Ser. Filos., Mosc. Univ. Math. Bull., 70, 2, 92-95, (March 2015), (in Russian); translation in:
[11] Mishchenko, S. P.; Valenti, A., An almost nilpotent variety of exponent 2, Isr. J. Math., 199, 241-258, (2014) · Zbl 1322.17001
[12] Mishchenko, S.; Valenti, A., On almost nilpotent varieties of subexponential growth, J. Algebra, 423, 902-915, (2015) · Zbl 1333.17001
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.