Algebra and Logic

, Volume 50, Issue 1, pp 46–61 | Cite as

Growth in Poisson algebras


A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x 1, y 1}, {x 2, y 2}, . . . , {x m , y m }} = 0 and {x 1, y 1} · {x 2, y 2} · . . . · {x m , y m } = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.


Poisson algebra growth of variety colength of variety 


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  1. 1.
    S. P. Mishchenko, V. M. Petrogradsky, and A. Regev, “Poisson PI algebras,” Trans. Am. Math. Soc., 359, No. 10, 4669–4694 (2007).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    I. P. Shestakov, “Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras,” Algebra Logika, 32, No. 5, 571–584 (1993).MATHMathSciNetGoogle Scholar
  3. 3.
    Yu. A. Bakhturin, Identities in Lie Algebras [in Russian], Nauka, Moscow (1985).Google Scholar
  4. 4.
    V. Drensky, Free Algebras and PI-Algebras. Graduate Course in Algebra, Springer, Singapore (2000).MATHGoogle Scholar
  5. 5.
    D. R. Farkas, “Poisson polynomial identities,” Comm. Alg., 26, No. 2, 401–416 (1998).CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    D. R. Farkas, “Poisson polynomial identities. II,” Arch. Math. (Basel), 72, No. 4, 252–260 (1999).MATHMathSciNetGoogle Scholar
  7. 7.
    S. M. Ratseev, “The growth and colength of spaces of a special form in varieties of Poisson algebras,” Izv. Vyssh. Uch. Zav. Povolzh. Reg., No. 5(26), 125–135 (2006).Google Scholar
  8. 8.
    Yu. N. Mal’tsev, “A basis for the identities of the algebra of upper triangular matrices,” Algebra Logika, 10, No. 4, 393–400 (1971).MATHMathSciNetGoogle Scholar
  9. 9.
    V. M. Petrogradsky, “On numerical characteristics of subvarieties for three varieties of Lie algebras,” Mat. Sb., 190, No. 6, 111–126 (1999).MathSciNetGoogle Scholar
  10. 10.
    V. M. Petrogradsky, “Exponents of subvarieties of upper triangular matrices over arbitrary fields are integral,” Serd. Math. J., 26, No. 2, 167–176 (2000).MATHMathSciNetGoogle Scholar
  11. 11.
    S. M. Ratseev, “The growth of some varieties of Leibniz algebras,” Vest. Samara State Univ., No. 6(46), 70–77 (2006).Google Scholar
  12. 12.
    S. P. Mishchenko, “Varieties of Lie algebras with weak growth of the sequence of codimensions,” Vest. Mosk. Univ., No. 5, 63–66 (1982).MathSciNetGoogle Scholar
  13. 13.
    S. M. Ratseev, “The growth of varieties of Leibniz algebras with nilpotent commutator subalgebra,” Mat. Zametki, 82, No. 1, 108–117 (2007).MathSciNetGoogle Scholar
  14. 14.
    A. R. Kemer, “T-ideals with polynomial growth of the codimensions are Specht,” Sib. Mat. Zh., 19, No. 1, 54–69 (1978).MATHMathSciNetGoogle Scholar
  15. 15.
    I. I. Benediktovich and A. E. Zalesskii, “T-ideals of free Lie algebras with polynomial growth of sequences of codimensions,” Izv. Akad. Nauk Bel. SSR, Ser. Fiz.-Mat. Nauk, No. 3, 5–10 (1980).Google Scholar
  16. 16.
    S. P. Mishchenko and O. I. Cherevatenko, “Varieties of Leibniz algebras with weak growth,” Vest. Samara State Univ., No. 9(49), 19–23 (2006).Google Scholar
  17. 17.
    S. P. Mishchenko, “Varieties of polynomial growth of Lie algebras over a field of characteristic zero,” Mat. Zametki, 40, No. 6, 713–721 (1986).MathSciNetGoogle Scholar

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© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Ul’yanovsk State UniversityUl’yanovskRussia

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