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Lie structure of smash products

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Abstract

We investigate the conditions under which the smash product of an (ordinary or restricted) enveloping algebra and a group algebra is Lie solvable or Lie nilpotent.

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References

  1. Yu. Bahturin and V. Petrogradsky, Polynomial identities in smash products, Journal of Lie Theory 12 (2002), 369–395.

    MathSciNet  MATH  Google Scholar 

  2. J. Bergen, D. M. Riley and H. Usefi, Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity, Israel Journal of Mathematics 196 (2013), 161–173.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. K. Bhandari and I. B. S. Passi, Lie nilpotency indices of group algebras, Bulletin of the London Mathematical Society 24 (1992), 68–70.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Bovdi and A. Grishkov, Lie properties of crossed products, Journal of Algebra 320 (2008), 3447–3460.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. A. Bovdi and I. I. Khripta, Generalized Lie nilpotent group rings, Mathematics of the U.S.S.R. Sbornik 57 (1987), 165–169.

    Article  MATH  Google Scholar 

  6. A. A. Bovdi and J. Kurdics, Lie properties of the group algebra and the nilpotency class of the group of units, Journal of Algebra 212 (1999), 28–64.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Dăscălescu, C. Năstăsescu and S. Raianu, Hopf Algebras. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 235, Marcel Dekker, New York, 2001.

    Google Scholar 

  8. N. D. Gupta and F. Levin, On the Lie ideals of a ring, Journal of Algebra 81 (1983), 225–231.

    Article  MathSciNet  MATH  Google Scholar 

  9. N. Jacobson, Lie Algebras, Dover, New York, 1979.

    MATH  Google Scholar 

  10. S. A. Jennings, Central chains of ideals in an associative ring, Duke Mathematical Journal 9 (1942), 341–355.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Kochetov, PI Hopf algebras of prime characteristic, Journal of Algebra 262 (2003), 77–98.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Montgomery, Hopf Algebras and their Actions on Rings, CMBS Regional Conference Series in Mathematics, Vol. 82, American Mathematical Society, Providence, RI, 1993.

    Book  MATH  Google Scholar 

  13. I. B. S. Passi, D. S. Passman and S. K. Sehgal, Lie solvable group rings, Canadian Journal of Mathematics 25 (1973), 748–757.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. M. Riley, PI-algebras generated by nilpotent elements of bounded index, Journal of Algebra 192 (1997), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. M. Riley and A. Shalev, The Lie structure of enveloping algebras, Journal of Algebra 162 (1993), 46–61.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. M. Riley and A. Shalev, Restricted Lie algebras and their envelopes, Canadian Journal of Mathematics 47 (1995), 146–164.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. M. Riley and V. Tasić, Lie identities for Hopf algebras, Journal of Pure and Applied Algebra 122 (1997), 127–134.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. K. Sehgal, Topics in Group Rings, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 50, Marcel Dekker, New York, 1978.

    MATH  Google Scholar 

  19. A. Shalev, The derived length of Lie soluble group rings. I, Journal of Pure and Applied Algebra 78 (1992), 291–300.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Shalev, The derived length of Lie soluble group rings. II, Journal of the London Mathematical Society 49 (1994), 93–99.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. K. Sharma and J. B. Srivastava, Lie solvable rings, Proceedings of the American Mathematical Society 94 (1985), 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Siciliano, Lie derived lengths of restricted universal enveloping algebras, Publicationes Mathematicae Debrecen 68 (2006), 503–513.

    MathSciNet  MATH  Google Scholar 

  23. S. Siciliano, On Lie solvable restricted enveloping algebras, Journal of Algebra 314 (2007), 226–234.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Siciliano and H. Usefi, Lie solvable enveloping algebras of characteristic two, Journal of Algebra 382 (2013), 314–331.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Strade, Simple Lie Algebras over Fields of Positive Characteristic I. Structure Theory, de Gruyter Expositions in Mathematics, Vol. 38, Walter de Gruyter & Co., Berlin, 2004.

    Book  MATH  Google Scholar 

  26. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 116, Marcel Dekker, New York, 1988.

    MATH  Google Scholar 

  27. H. Usefi, Lie identities on enveloping algebras of restricted Lie superalgebras, Journal of Algebra 393 (2013), 120–131.

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Zalesskiĭ and M. B. Smirnov, Associative rings satisfying the identity of Lie solvability, Vestsī Akadèmīī Navuk BSSR. Seryya Fīzīka-Matèmatychnykh Navuk 2 (1982), 15–20.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via Provinciale Lecce–Arnesano, 73100, Lecce, Italy

    Salvatore Siciliano

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada, A1C 5S7

    Hamid Usefi

Authors
  1. Salvatore Siciliano
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  2. Hamid Usefi
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Corresponding author

Correspondence to Salvatore Siciliano.

Additional information

The research of the second author was supported by NSERC of Canada under grant # RGPIN 418201.

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Siciliano, S., Usefi, H. Lie structure of smash products. Isr. J. Math. 217, 93–110 (2017). https://doi.org/10.1007/s11856-017-1439-5

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  • DOI: https://doi.org/10.1007/s11856-017-1439-5

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