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On the codimension growth of almost nilpotent Lie algebras

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Abstract

We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra L is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of L exists and is a positive integer.

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References

  1. Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, Utrecht 1987.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Giambruno, A. Regev and M. Zaicev, On the codimension growth of finite-dimensional Lie algebras, Journal of Algebra 220 (1999), 466–474.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Giambruno, A. Regev and M. V. Zaicev, Simple and semisimple Lie algebras and codimension growth, Transactions of the American Mathematical Society 352 (2000), 1935–1946.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  10. S. P. Mishchenko, On the problem of the Engel property (Russian), Rossiıskaya Akademiya Nauk. Matematicheskiı Sbornik 124 (166) (1984), 56–67.

    MathSciNet  Google Scholar 

  11. S. P. Mishchenko, Growth of varieties of Lie algebras (Russian), Uspekhi Mat. Nauk 45 (1990), 25–45; English translation: Russian Math. Surveys 45 (1990), 27–52.

    MathSciNet  Google Scholar 

  12. S. P. Mishchenko and P. M. Petrogradsky, Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra, Communications in Algebra 27 (1999), 2223–2230.

    Article  MathSciNet  MATH  Google Scholar 

  13. V. M. Petrogradsky, Scale for codimension growth of Lie algebras, in Methods in Lie Theory (Levico Terme, 1997), Lecture Notes in Pure and Applied Mathematics 198, Marcel Dekker, New York, 1998, pp. 213–222.

    Google Scholar 

  14. Yu. P. Razmyslov, Identities of Algebras and their Representations, Translations of Mathematical Monographs 138, American Mathematical Society, Providence, RI, 1994.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. A. B. Verevkin, M. V. Zaicev and S. P. Mishchenko, A sufficient condition for coinciding of lower and upper exponents of a variety of linear algebras (Russian), Moskovskogo Universiteta, Seriya I, Matematika, Mekhanika, to appear.

  17. M. V. Zaicev, Standard identity in special Lie algebras (Russian), Moskovskogo Universiteta. Seriya I. Matematika, Mekhanika no. 1, (1993), 56–59.

  18. M. Zaicev, Identities of affine Kac-Moody algebras (Russian), Moskovskogo Universiteta, Seriya I, Matematika, Mekhanika no. 2 (1996), 33–36; English translation: Moscow University Mathematics Bulletin 51 (1996), 29–31.

  19. M. Zaicev, Varieties of affine Kac-Moody algebras (Russian), Rossiıskaya Akademiya Nauk, Matematicheskie Zametki 62 (1997), 95–102.

    Google Scholar 

  20. M. Zaicev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras (Russian), Rossiıskaya Akademiya Nauk, Izvestiya, Seriya Matematicheskaya 66 (2002), 23–48; English translation: Izvestiya Mathematics 66 (2002), 463–487.

    MathSciNet  Google Scholar 

  21. M. V. Zaicev and S. P. Mishchenko, An example of a variety of Lie algebras with a fractional exponent, Journal of Mathematical Sciences 93 (1999), 977–982.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. V. Zaicev and S. P. Mishchenko, On the polynomial growth of the colength of varieties of Lie algebras (Russian), Algebra i Logika 38 (1999), 161–175; English translation: Algebra and Logic 38 (1999), 84–92.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P. O. B. 2964, Ljubljana, 1001, Slovenia

    Dušan Repovš

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

    Mikhail Zaicev

Authors
  1. Dušan Repovš
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  2. Mikhail Zaicev
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Corresponding author

Correspondence to Dušan Repovš.

Additional information

The first author was partially supported by the Slovenian Research Agency grants P1-0292-0101-04 and J1-2057-0101.

The second author was partially supported by RFBR grant No 09-01-00303a.

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Repovš, D., Zaicev, M. On the codimension growth of almost nilpotent Lie algebras. Isr. J. Math. 194, 137–150 (2013). https://doi.org/10.1007/s11856-012-0120-2

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  • DOI: https://doi.org/10.1007/s11856-012-0120-2

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