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Codimension growth of simple Jordan superalgebras

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Abstract

We study asymptotic behaviour of graded and non-graded codimensions of simple Jordan superalgebras over a field of characteristic zero. It is known that the PI-exponent of any finite-dimensional associative or Jordan or Lie algebra A is a non-negative integer less than or equal to the dimension of algebra A. Moreover, the PI-exponent is equal to the dimension if and only if A is simple provided that the base field is algebraically closed. In the present paper we prove that for a Jordan superalgebra P(t) = H(Mt∣t, trp) its non-graded and ℤ2-graded exponents are strictly less than dim P(t). In particular, exp P(2) is fractional.

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References

  1. Yu. A. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, Utrecht, 1987.

    MATH  Google Scholar 

  2. Yu. Bahturin and V. Drensky, Graded polynomial identities of matrices, Linear Algebra and its Applications 357 (2002), 15–34.

    Article  MathSciNet  Google Scholar 

  3. A. Berele, Properties of hook Schur functions with applications to p.i. algebras, Advances in Applied Mathematics 41 (2008), 52–75.

    Article  MathSciNet  Google Scholar 

  4. A. Berele and A. Regev, Asymptotic behaviour of codimensions of p.i. algebras satisfying Capelli identities, Transactions of the American Mathematical Society 360 (2008), 5155–5172.

    Article  MathSciNet  Google Scholar 

  5. V. Drensky, Codimensions of T-ideals and Hilbert series of relatively free algebras, Journal of Algebra 91 (1984), 1–17.

    Article  MathSciNet  Google Scholar 

  6. V. Drensky, Polynomial identities for the Jordan algebra of a symmetric bilinear form, Journal of Algebra 108 (1987), 66–87.

    Article  MathSciNet  Google Scholar 

  7. V. Drensky, Free algebras and PI-algebras, Springer, Singapore, 2000.

    MATH  Google Scholar 

  8. A. S. Dzhumadil’daev, Codimension growth and non-Koszulity of Novikov operad, Communications in Algebra 39 (2011), 2943–2952.

    Article  MathSciNet  Google Scholar 

  9. A. Giambruno, I. Shestakov and M. Zaicev, Finite-dimensional non-associative algebras and codimension growth, Advances in Applied Mathematics 47 (2011), 125–139.

    Article  MathSciNet  Google Scholar 

  10. A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Advances in Mathematics 140 (1998), 145–155.

    Article  MathSciNet  Google Scholar 

  11. A. Giambruno and M. Zaicev, Exponential codimension growth of PI algebras: an exact estimate, Advances in Mathematics 142 (1999), 221–243.

    Article  MathSciNet  Google Scholar 

  12. A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Mathematical Surveys and Monographs, Vol. 122, American Mathematical Society, Providence, RI, 2005.

    Book  Google Scholar 

  13. A. Giambruno and M. Zaicev, Codimension growth of special simple Jordan algebras, Transactions of the American Mathematical Society 362 (2010), 3107–3123.

    Article  MathSciNet  Google Scholar 

  14. A. Giambruno and M. Zaicev, On codimension growth of finite-dimensional Lie superalgebras, Journal of the London Mathematical Society 85 (2012), 534–548.

    Article  MathSciNet  Google Scholar 

  15. A. Giambruno and M. Zaicev, Sturmian words and overexponential codimension growth, Advances in Applied Mathematics 95 (2018), 53–64.

    Article  MathSciNet  Google Scholar 

  16. A. Giambruno and E. Zelmanov, On growth of codimensions of Jordan algebras, in Groups, Algebras and Applications, Contemporary Mathematics, Vol. 537, American Mathematical Society, Providence, RI, 2011, pp. 205–210.

    Chapter  Google Scholar 

  17. A, N. Grishkov, Growth of varieties of Lie algebras, Matematicheskie Zametki 44 (1988), 51–54, 154; English translation: Mathematical Notes 44 (1988), 515–517.

    MathSciNet  MATH  Google Scholar 

  18. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, Vol. 16. Addison-Wesley, Reading, MA, 1981.

    MATH  Google Scholar 

  19. S. P. Mishchenko, Growth of varieties of Lie algebras, Uspekhi Matematicheskikh Nauk 45 (1990), 25–45, 189; English translation: Russian Mathematical Surveys 45 (1990), 27–52

    MathSciNet  Google Scholar 

  20. V. M. Petrogradskii, On types of superexponential growth of identities in Lie PI-algebras, Fundamenatal’naya i Prikladnaya Matematika 1 (1995), 989–1007.

    MathSciNet  MATH  Google Scholar 

  21. M. L. Racine and E. I. Zelmanov, Simple Jordan superalgebras with semisimple even part, Journal of Algebra 270 (2003), 374–444.

    Article  MathSciNet  Google Scholar 

  22. A. Regev, Existence of identities in A ⊗ B, Israel Journal of Mathematics 11 (1972), 131–152.

    Article  MathSciNet  Google Scholar 

  23. D. Repovš and M. Zaicev, Graded identities of some simple Lie superalgebras, Algebras and Representation Theory 17 (2014), 1401–1412.

    Article  MathSciNet  Google Scholar 

  24. D. Repovš and M. Zaicev, Graded codimensions of Lie superalgebra b(2), Journal of Algebra 422 (2015), 1–10.

    Article  MathSciNet  Google Scholar 

  25. D. Repovš and M. Zaicev, Graded PI-exponents of simple Lie superalgebras, Arkiv för Matematik 54 (2016), 147–156.

    Article  MathSciNet  Google Scholar 

  26. D. Repovš and M. Zaicev, Identities of graded simple algebras, Linear and Multilinear Algebra 65 (2017), 44–57.

    Article  MathSciNet  Google Scholar 

  27. I. B. Volichenko, Varieties of Lie algebras with identity [[X1, X2, X3], [X4, X5, X6]] = 0 over a field of characteristic zero, Sibirskiĭ MatematicheskiĭZhurnal 25 (1984), 40–54.

    MathSciNet  Google Scholar 

  28. M. V. Zaitsev, Varieties of affine Kac-Moody algebras, Matematicheskie Zametki 62 (1997), 95–102; English translation: Mathematical Notes 62 (1997), 80–86.

    Article  MathSciNet  Google Scholar 

  29. M V. Zaicev, Graded identities in finite-dimensional algebras of codimensions of identities in associative algebras, Moscow University Mathematics Bulletin 70 (2015), 234–236.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Instituto De Matemática E Estatística, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-090, São Paulo, Brazil

    Ivan Shestakov

  2. Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia

    Ivan Shestakov

  3. Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992, Russia

    Mikhail Zaicev

Authors
  1. Ivan Shestakov
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  2. Mikhail Zaicev
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Correspondence to Mikhail Zaicev.

Additional information

The first author was supported by the CNPq grant 304313/2019-0 and FAPESP grant 2018/23690-6.

The second author was partially supported by FAPESP grant 2019/02510-2 and by Russian Science Foundation, grant 16-11-10013-P.

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Shestakov, I., Zaicev, M. Codimension growth of simple Jordan superalgebras. Isr. J. Math. 245, 615–638 (2021). https://doi.org/10.1007/s11856-021-2221-2

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  • DOI: https://doi.org/10.1007/s11856-021-2221-2

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