NUMERICAL CHARACTERISTICS OF LEIBNIZ-POISSON ALGEBRAS
https://doi.org/10.22405/2226-8383-2017-18-1-143-159
Abstract
The paper is survey of recent results of investigations on varieties of Leibniz-Poisson algebras. We show that a variety of Leibniz-Poisson algebras has either polynomial growth or growth with exponential not less than 2, the eld being arbitrary. We show that every variety of Leibniz-Poisson algebras of polynomial growth over a eld of characteristic zero has a nite basis for its polynomial identities.We construct a variety of Leibniz-Poisson algebras with almost polynomial growth. We give equivalent conditions of the polynomial codimension growth of a variety of Leibniz-Poisson algebras over a eld of characteristic zero. We show all varieties of Leibniz-Poisson algebras with almost polynomial growth in one class of varieties. We study varieties of Leibniz-Poisson algebras, whose ideals of identities contain the identity fx; yg fz; tg = 0, we study an interrelation between such varieties and varieties of Leibniz algebras. We show that from any Leibniz algebra L one can construct the Leibniz-Poisson algebra A and the properties of L are close to the properties of A. We show that if the ideal of identities of a Leibniz-Poisson variety V does not contain any Leibniz polynomial identity then V has overexponential growth of the codimensions. We construct a variety of Leibniz-Poisson algebras with almost exponential growth. Let f n(V)gn1 be the sequence of proper codimension growth of a variety of Leibniz-Poisson algebras V. We give one class of minimal varieties of Leibniz-Poisson algebras of polynomial growth of the sequence f n(V)gn1, i.e. the sequence of proper codimensions of any such variety grows as a polynomial of some degree k, but the sequence of proper codimensions of any proper subvariety grows as a polynomial of degree strictly less than k.
About the Authors
S. M. Ratseev
Ulyanovsk State University
Russian Federation
Russian Federation
Doctor of Physics and Mathematics, Associate Professor, Professor at the Department of Information Security and Control Theory
O. I. Cherevatenko
Ulyanovsk State I. N. Ulyanov Pedagogical University
Russian Federation
Russian Federation
Associate Professor, Dept. of Higher Mathematics
References
1. Ratseev S. M. 2012, ''Commutative Leibniz-Poisson algebras of polynomial growth'', Vestnik SamGU. Estestvenno-Nauchnaya Ser., vol. 94, no 3/1, pp. 54--65.
2. Bahturin Yu. A. 1987, ''Identical relations in Lie algebras'', VNU Science Press, Utrecht.
3. Giambruno A., Zaicev M. V. 2005, ''Polynomial Identities and Asymptotic Methods'', AMS Mathematical Surveys and Monographs. Vol. 122. Providence R. I.
4. Drensky, V. 2000, ''Free algebras and PI-algebras. Graduate course in algebra'', Singapore: Springer-Verlag.
5. Regev A. 1971, ''Existence of polynomial identities in $A\otimes B$'', Bull. Amer. Math. Soc., vol. 77, no 6, pp. 1067--1069.
6. Giambruno A., Zaicev M. V. 1999, ''Exponential codimension growth of P. I. algebras: an exact estimate'', Adv. Math., vol. 142, pp. 221--243.
7. Kemer A. R. 1978, ''T-ideals with power growth of the codimensions are Specht'', Sibirsk. Mat. Jh., vol. 19, no. 1, pp. 54--69. English translation: Siberian Math. J., 1978, vol. 19, no. 1. pp. 37--48.
8. Drensky V., Regev A. 1996, ''Exact behaviour of the codimention of some P. I. algebras'', Israel J. Math., vol. 96, pp. 231--242.
9. Volichenko I. B. 1984, ''Varieties of Lie algebras with identity $[[X_1,X_2,X_3],[X_4,X_5,X_6]]$ = $0$ over a field of characteristic zero'', Sibirsk. Mat. Jh., vol. 25, no. 3, pp. 40--54. English translation: Siberian Math. J., 1984, vol. 25, pp. 370--382.
10. Petrogradsky, V. M. 1997, ''Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions'', Mat. Sb., vol. 188, no. 6, pp. 119--138. English translation: Russian Acad. Sci. Sb. Math., 1997, vol. 188, no. 6, pp. 913--931.
11. Ratseev S. M. 2012, ''Equivalent conditions of polynomial growth of a variety of Poisson algebras'', Moscow University Mathematics Bulletin, vol. 67, no 5-6, pp. 195--199. (in Eng. )
12. Ratseev S. M. 2014, ''Necessary and sufficient conditions of polynomial growth of varieties of Leibniz-Poisson algebras'', Russian Mathematics (Iz. VUZ. ), no 3, pp. 26--30. (in Eng. )
13. Ratseev S. M. 2015, ''Lie algebras with extremal properties'', Sibirsk. Mat. Jh., vol. 56, no 2, pp. 444--454. English translation: Siberian Math. J., 2015, vol. 56, no 2, pp. 358--366.
14. Petrogradsky, V. M. 1999, ''On numerical characteristics of subvarities of three varieties of Lie algebras'', Mat. Sb., vol. 190, no. 6, pp. 111--126. English translation: Sb. Math., vol. 190, no. 6, pp. 913--931.
15. Petrogradsky V. M. 2000, ''Exponents of subvarieties of upper triangular matrices over arbitrary fields are integral'', Serdica Math. J., vol. 26, no. 2, pp. 167--176.
16. Ratseev S. M. 2011, ''Identities in the varieties generated by the algebras of upper triangular matrices'', Siberian Mathematical Journal, vol. 52, no 2, pp. 329--339. (in Eng. )
17. Ratseev S. M. 2006, ''Growth of some varieties of leibniz algebras'', Vestnik SamGU. Estestvenno-Nauchnaya Ser., vol. 46, no 6/1. pp. 70--77.
18. Ratseev S. M. 2010, ''Asymptotic behavior of the codimentions growth of Leibniz algebras with a nilpotent commutator subalgebra'', Vestnik SamGU. Estestvenno-Nauchnaya Ser., vol. 78, no 4, pp. 65--72.
19. Ratseev S. M. 2013, ''On exponents of some varieties of linear algebras'' Prikl. Diskr. Mat., vol. 21, no 3, pp. 32--34.
20. Ratseev S. M. 2011, ''Growth of some varieties of Leibniz-Poisson algebras'', Serdica Mathematical Journal, vol. 37, no 4, pp. 331--340.
21. Ratseev S. M., Cherevatenko O. I. 2013, ''Exponents of some varieties of Leibniz-Poisson algebras'', Vestnik SamGU. Estestvenno-Nauchnaya Ser., vol. 104, no 3, pp. 42--52.
22. Ratseev S. M. 2013, ''On varieties of Leibniz-Poisson algebras with the identity $\{x, y\}\cdot \{z, t\}=0$'', J. Sib. Fed. Univ. Math. Phys., vol. 6, no 1, pp. 97--104.
23. Skoraya T. V., Frolova Yu. Yu. 2014, ''About variety ${}_3{\bf N}$ of Leibniz algebras and its subvarieties'', Chebyshevskii Sb., vol. 15, no 1, pp. 155--185.
24. Ratseev S. M., Cherevatenko O. I. 2012, ''On the nilpotent Leibniz-Poisson algebras'', Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz. -Mat. Nauki., vol. 29, no 4, pp. 207--211.
25. Ratseev S. M., Cherevatenko O. I. 2015, ''Complexity functions of some Leibniz-Poisson algebras'', Sib. Elektron. Mat. Izv. Vol. 12, pp. 500--507.
26. Mishchenko S., Valenti A. 2005 ''A Leibniz variety with almost polynomial growth'', J. Pure Appl. Algebra., vol. 202, no 1-3, pp. 82--101.
27. Abanina L. E., Ratseev S. M. 2005, ''Variety of Leibniz algebras connected with standard identities'', Vestnik SamGU. Estestvenno-Nauchnaya Ser., vol. 40, no 6, pp. 36--50.
28. Abanina, L. E., Mishchenko S. P. 2002, ''Some Leibniz varieteis'', Mathematical methods and applications. Proceedings of the 14th Mathematical Readings, pp. 95--99.
29. Ratseev S. M., Cherevatenko O. I. 2013, ''On some varieties of Leibniz-Poisson algebras with extreme properties'', Vestn. Tomsk. Gos. Univ. Mat. Mekh., vol. 22, no 2, pp. 57--59.
30. Giambruno A., Mattina D. La., Petrogradsky V. M. 2007, ''Matrix algebras of polynomial codimention growth'', Israel J. Math., vol. 158, pp. 367--378.
31. Ratseev S. M. 2014, ''On minimal Leibniz-Poisson algebras of polynomial growth'', Far Eastern Mathematical Journal, vol. 14, no 2, pp. 248--256.
Review
For citations:
Ratseev S.M., Cherevatenko O.I. NUMERICAL CHARACTERISTICS OF LEIBNIZ-POISSON ALGEBRAS. Chebyshevskii Sbornik. 2017;18(1):143-159. (In Russ.) https://doi.org/10.22405/2226-8383-2017-18-1-143-159
Views:
875
Email this article 