Variety with fractional codimension growth and the Specht problem
https://doi.org/10.22405/2226-8383-2018-19-1-176-186
Abstract
According to A.I. Maltsev, a set of linear algebras in which a fixed set of identities is called a variety. Using the language of the theory of Lie algebras, we say that the algebra is metabelian if it satisfies the identity (xy)(zt) ≡ 0. A variety is called Specht if it is such a variety and any of its subvariety has a finite basis of identities. Codimension growth is determined by sequence of dimensions multilinear parts of a relatively free algebra of a variety. This sequence is called a sequence codimensions, referring to the multilinear spaces of the ideal identities of the variety. This article presents the results related to the problem of fractional polynomial growth. The review gives new examples of such varieties, and also give a new example of a variety with an infinite basis of identities.
About the Authors
S. P. Mishchenko
Ulyanovsk state University
Russian Federation
Russian Federation
Mishchenko Sergey Petrovich — doctor of physical and mathematical sciences, professor, professor of the department of applied mathematics
O. V. Shulezhko
Ulyanovsk state pedagogical University named after I.N. Ulyanov
Russian Federation
Russian Federation
Shulezhko Olesya Vladimirovna — candidate of physics-mathematical sciences, associate professor, deputy dean for educational work of the faculty Fmita
References
1. Giambruno, A., Zaicev, M. 2005, “Polynomial Identities and Asymptotic Methods”, Mathematical Surveys and Monographs, AMS, Providence, RI, vol. 122, 352 p.
2. Mal’tsev, A.I. 1950. “On algebras defined by identities,” Mat. Sb., vol. 26(68), issue 1, pp. 19–33. (in Russian)
3. Drensky, V. 1992, ”Relations for the cocharacter sequences of T-ideals”, Proc. of the International Conference on Algebra Honoring A. Malcev, Contemp. Math., vol.131, part 2, pp. 285–300.
4. Mishchenko, S., Valenti, A. 2016, ”Codimension and colength sequences of algebras and growth phenomena”,Sao Paulo Journal of Mathematical Sciences, vol. 10 , issue 2, pp. 263–272. Article First Online: 30 November 2015 DOI: 10.1007/s40863-015-0025-1
5. Giambruno, A., Mishchenko, S., Zaicev, M. 2006, ”Algebras with intermediate growth of the codimensions”, Adv. in Appl. Math. 37, no. 3, pp.360 – 377.
6. Mishchenko, S.P., Zaicev, M.V. 1999, ”An example of a variety of Lie algebras with a fractional exponent”, Journal of Mathematical Sciences (New York), vol. 93,no 6, pp. 977–982.
7. Malyusheva,O., Mishchenko, S., Verevkin, A. 2013, ”Series of varieties of Lie algebras of different fractional exponents”, Compt. rend. Acad. Bulg. Sci., 66, no 3, pp. 321–330.
8. Bogdanchuk, O.A., Mishchenko, S.P., Verëvkin,A.B. 2014, ”On Lie algebras with exponential growth of the codimensions”, Serdica Math. J., vol. 40, no 3-4, pp. 209–240.
9. Mishchenko, S.S., 2011, "New example of a variety of lie algebras with fractional exponent", Vestnik Moskov. Univ. Ser. I Mat. Mekh., no 6, pp. 44–47; English translation in: Moscow University Mathematics Bulletin, vol. 66, no 6, pp. 264–266.
10. Giambruno, A., Mishchenko, S., Zaicev, M. 2008, ”Codimensions of Algebras and Growth Functions”, Advances in Mathematics, vol. 217, no 3, pp. 1027–1052.
11. Zaicev, M. 2014, ”On existence of PI-exponents of codimension growth”, Electron. Res. Announc. Math. Sci., vol.21, pp. 113–119.
12. Zaicev, M., Mishchenko, S. 2008, ”The example of linear algebras variety with fractional polynomial growth”, Vestn. Mosk. Univ., ser. I , vol.1, pp. 25–31. (in Russian)
13. Mishchenko,S.P.2013,”Theexampleoflinearalgebrasvarietywithfractionalpolynomialgrowth less than 3”, Vestn. Mosk. Univ., ser. I , no 3, pp. 51–54. (in Russian)
14. Giambruno, A., Mishchenko S., Valenti, A., Zaicev, M. 2017, ”Polynomial codimension growth and Specht problem”, Journal of Algebra, no. 469 , pp. 421-436.
15. Giambruno, A., Mishchenko, S.P. 2010, ”Polynomial growth of the codimensions: A characOfitserovterization”, Proc. Amer. Math. Soc., vol. 138, no 3, pp. 853–859.
16. Mishchenko, S.P., Verevkin, A.B. 2016, ”On varieties with identities of one generated free metabelian algebra”, Chebyshevskii sbornik, vol. 17, no. 2 (58), pp. 21–55. (in Russian)
17. Drenski, V.S. 1982, ”Representations of the symmetric group and varieties of linear algebras”, Math. USSR Sbornik, vol. 43, no. 1, pp. 85–101.
Review
For citations:
Mishchenko S.P., Shulezhko O.V. Variety with fractional codimension growth and the Specht problem. Chebyshevskii Sbornik. 2018;19(1):176-186. (In Russ.) https://doi.org/10.22405/2226-8383-2018-19-1-176-186
Views:
596
Email this article 