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Numerical Characteristics of Varieties of Poisson Algebras

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This paper is a survey of recent results of investigations on varieties of Poisson algebras. We give constructions of varieties of Poisson algebras with extremal properties, we give equivalent conditions for the polynomial codimension growth of a variety of Poisson algebras, we study varieties of Poisson algebras whose ideals of identities contain the identity {x, y} ⋅ {z, t} = 0, and we study the interrelation between such varieties and varieties of Lie algebras.

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References

  1. Yu. A. Bahturin, Identical Relations in Lie Algebras, VNU Sci. Press, Utrecht (1987).

  2. O. I. Cherevatenko, “On Lie nilpotent Poisson algebras,” Nauch. Ved. BelGU. Mat., Fiz., 142, No. 23, 14–16 (2012).

    Google Scholar 

  3. V. Drensky, “Representations of the symmetric group and varieties of linear algebras,” Math. USSR. Sb., 43, 85–100 (1981).

    Article  Google Scholar 

  4. V. Drensky, “Relations for the cocharacter sequences of T-ideals,” in: Proc. of the Int. Conf. on Algebra Honoring A. Malcev, Contemp. Math., Vol. 131, Pt. 2, Amer. Math. Soc., Providence (1992), pp. 285–300.

  5. V. Drensky, Free Algebras and PI-Algebras. Graduate Course in Algebra, Springer, Singapore (2000).

  6. V. Drensky and A. Regev, “Exact asymptotic behaviour of the codimensions of some P.I. algebras,” Israel J. Math., 96, 231–242 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  7. D. R. Farkas, “Poisson polynomial identities,” Commun. Algebra, 26, No. 2, 401–416 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. R. Farkas, “Poisson polynomial identities. II,” Arch. Math. (Basel), 72, No. 4, 252–260 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Giambruno, D. La Mattina, and V. M. Petrogradsky, “Matrix algebras of polynomial codimension growth,” Israel J. Math., 158, 367–378 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Giambruno and M. V. Zaicev, “Exponential codimension growth of P.I. algebras: an exact estimate,” Adv. Math., 142, 221–243 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Giambruno and M. V. Zaicev, Polynomial Identities and Asymptotic Methods, Math. Surv. Monographs, Vol. 122, Amer. Math. Soc., Providence (2005).

  12. A. R. Kemer, “T-ideals with power growth of the codimensions are Specht,” Sib. Math. J., 19, No. 1, 37–48 (1978).

    Article  MATH  Google Scholar 

  13. D. La Mattina, “Varieties of almost polynomial growth: classifying their subvarieties,” Manuscripta Math., 123, No. 2, 185–203 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. P. Mishchenko, “Varieties of Lie algebras with two-step nilpotent commutant,” Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, No. 6, 39–43 (1987).

  15. S. P. Mishchenko, “On varieties of solvable Lie algebras,” Sov. Math. Dokl., 42, No. 1, 202–205 (1991).

    MathSciNet  MATH  Google Scholar 

  16. S. P. Mishchenko, V. M. Petrogradsky, and A. Regev, “Poisson PI algebras,” Trans. Am. Math. Soc., 359, No. 10, 4669–4694 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  17. S. P. Mishchenko and M. V. Zaicev, “An example of a variety of Lie algebras with a fractional exponent,” J. Math. Sci., 93, No. 6, 977–982 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. M. Petrogradsky, “Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions,” Russ. Acad. Sci. Sb. Math. 188, No. 6, 913–931 (1997).

    MATH  Google Scholar 

  19. V. M. Petrogradsky, “On numerical characteristics of subvarieties of three varieties of Lie algebras,” Sb. Math., 190, No. 6, 913–931 (1999).

    MathSciNet  Google Scholar 

  20. V. M. Petrogradsky, “Exponents of subvarieties of upper triangular matrices over arbitrary fields are integral,” Serdica Math. J., 26, No. 2, 167–176 (2000).

    MathSciNet  MATH  Google Scholar 

  21. S. M. Ratseev, “A growth and a colength of spaces of a special type of varieties of Poisson algebras,” Izv. Vyssh. Uchebn. Zaved. Povolzh. Reg., 26, No. 5, 125–135 (2006).

    Google Scholar 

  22. S. M. Ratseev, “Growth of some varieties of Leibniz algebras,” Vestn. Samar. Gos. Univ. Estestv.-Nauch. Ser., 46, No. 6/1, 70–77 (2006).

    MathSciNet  MATH  Google Scholar 

  23. S. M. Ratseev, “Asymptotic behavior of the codimension growth of Leibniz algebras with a nilpotent commutator subalgebra,” Vestn. Samar. Gos. Univ. Estestv.-Nauch. Ser., 78, No. 4, 65–72 (2010).

    Google Scholar 

  24. S. M. Ratseev, “Growth in Poisson algebras,” Algebra Logic, 50, No. 1, 46–61 (2011).

    MathSciNet  MATH  Google Scholar 

  25. S. M. Ratseev, “Identities in the varieties generated by the algebras of upper triangular matrices,” Sib. Math. J., 52, No. 2, 329–339 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. M. Ratseev, “Equivalent conditions of polynomial growth of a variety of Poisson algebras,” Moscow Univ. Math. Bull., 67, No. 5-6, 195–199 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  27. S. M. Ratseev, “On varieties of Poisson algebras with extremal properties,” Nauch. Vedom. BelGU. Mat., Fiz., No. 5, 107–110 (2013).

  28. S. M. Ratseev, “Poisson algebras of polynomial growth,” Sib. Math. J., 54, No. 3, 555–565 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  29. S. M. Ratseev, “The interrelation of Poisson algebras and Lie algebras in the language of identities,” Mat. Zametki, 96, No. 4, 567–577 (2014).

    Article  MathSciNet  Google Scholar 

  30. S. M. Ratseev, “Lie algebras with extremal properties,” Sib. Math. J., 56, No. 2, 358–366 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  31. S. M. Ratseev, “On minimal Poisson algebras,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 11, 64–72 (2015).

  32. S. M. Ratseev, “On proper T-ideals of Poisson algebras with extreme properties,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 8–16 (2016).

  33. S. M. Ratseev and O. I. Cherevatenko, “Exponents of some varieties of Leibniz-Poisson algebras,” Vestn. Samar. Gos. Univ. Estestv.-Nauch. Ser., 104, No. 3, 42–52 (2013).

    MATH  Google Scholar 

  34. Yu. P. Razmyslov, Identities of Algebras and Their Representations, Transl. Math. Monogr., Vol. 138, Amer. Math. Soc., Providence (1994).

  35. A. Regev, “Existence of polynomial identities in A⊗B,” Bull. Am. Math. Soc., 77, No. 6, 1067–1069 (1971).

  36. I. P. Shestakov, “Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras,” Algebra Logic, 32, No. 5, 309–317 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  37. A. B. Verevkin, M. V. Zaicev, and S. P. Mishchenko, “A sufficient condition for existence of lower and upper exponents of the variety of linear algebras,” Moscow Univ. Math. Bull., No. 2, 86–89 (2011).

  38. I. B. Volichenko, “A variety of Lie algebras connected with standard identities,” Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, No. 1, 23–30 (1980).

  39. I. B. Volichenko, “A variety of Lie algebras connected with standard identities,” Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, No. 2, 22–27 (1980).

  40. I. B. Volichenko, “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,” Sib. Math. J., 25, 370–382 (1984).

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  1. Ulyanovsk State University, Ulyanovsk, Russia

    S. M. Ratseev

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Correspondence to S. M. Ratseev.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 2, pp. 217–242, 2016.

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Ratseev, S.M. Numerical Characteristics of Varieties of Poisson Algebras. J Math Sci 237, 304–322 (2019). https://doi.org/10.1007/s10958-019-4157-1

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