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Simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0

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Abstract

We classify the central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0. As a corollary, we describe the Poisson brackets on the simple finite-dimensional Jordan superalgebras of characteristic 0.

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References

  1. Pozhidaev A. P. and Shestakov I. P., “Noncommutative Jordan superalgebras of degree n > 2,” Algebra and Logic, 49, No. 1, 18–42 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  2. Schafer R. D., “Noncommutative Jordan algebras of characteristic 0,” Proc. Amer. Math. Soc., 6, 472–475 (1955).

    MathSciNet  MATH  Google Scholar 

  3. Oehmke R. H., “On flexible algebras,” Ann. of Math. (2), 68, No. 2, 221–230 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  4. McCrimmon K., “Structure and representations of noncommutative Jordan algebras,” Trans. Amer. Math. Soc., 121, 187–199 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  5. McCrimmon K., “Noncommutative Jordan rings,” Trans. Amer. Math. Soc., 158, No. 1, 1–33 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  6. Smith K. C., “Noncommutative Jordan algebras of capacity two,” Trans. Amer. Math. Soc., 158, No. 1, 151–159 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  7. Kokoris L. A., “Nodal non-commutative Jordan algebras,” Canad. J. Math., 12, 488–492 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  8. Kokoris L. A., “Simple nodal noncommutative Jordan algebras,” Proc. Amer. Math. Soc., 9, No. 4, 652–654 (1958).

    MathSciNet  MATH  Google Scholar 

  9. Shestakov I. P., “Certain classes of noncommutative Jordan rings,” Algebra and Logic, 10, No. 4, 252–280 (1971).

    Article  MATH  Google Scholar 

  10. Skosyrskiĭ V. G., Strongly prime noncommutative Jordan algebras in: Studies on the Theory of Rings and Algebras [in Russian], Trudy Inst. Mat. (Novosibirsk), 1989, 16, pp. 131–163.

    MathSciNet  Google Scholar 

  11. Kac V., “Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras,” Comm. Algebra, 13, No. 5, 1375–1400 (1977).

    Article  Google Scholar 

  12. Kantor I. L., “Jordan and Lie superalgebras determined by a Poisson algebra,” in: Second Siberian Winter School “Algebra and Analysis.” Proceedings of the Second Siberian School, Tomsk State University, Tomsk, Russia, 1989, Amer. Math. Soc., Providence, RI, 1992, pp. 55–80.

    Google Scholar 

  13. Kaplansky I., “Superalgebras,” Pacific J. Math., 80, 93–98 (1980).

    Article  MathSciNet  Google Scholar 

  14. Racine M. and Zelmanov E., “Simple Jordan superalgebras with semisimple even part,” J. Algebra, 270, No. 2, 374–444 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  15. Martinez C. and Zelmanov E., “Simple finite-dimensional Jordan superalgebras of prime characteristic,” J. Algebra, 236, No. 2, 575–629 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  16. Zelmanov E., “Semisimple finite dimensional Jordan superalgebras,” in: Lie Algebras, Rings and Related Topics. Papers of the 2nd Tainan-Moscow International Algebra Workshop’97, Tainan, Taiwan, January 11–17, 1997, Springer-Verlag, Hong Kong, 2000, pp. 227–243.

    Google Scholar 

  17. Filippov V. T., Kharchenko V. K., and Shestakov I. P., Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules, Inst. Math., Novosibirsk (1993).

    Google Scholar 

  18. Sabinin L. V., Sbitneva L., and Shestakov I., “Non-associative algebra and its applications,” in: Proc. 5th Intern. Conf., Oaxtep, Mexico, July 27–August 2, 2003, p. 516 (Lecture Notes Pure Appl. Math.; V. 246).

  19. Albert A. A., “Power-associative rings,” Trans. Amer. Math. Soc., 64, 552–593 (1948).

    Article  MathSciNet  MATH  Google Scholar 

  20. Dedkov A. I., “Some properties of quasiassociative and quasialternative algebras,” Siberian Math. J., 30, No. 3, 479–483 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  21. Shestakov I. P., “Alternative and Jordan superalgebras,” in: 10 Siberian School on Algebra, Geometry, andMathematical Physics, Inst. Mat., Novosibirsk, 1997, pp. 157–169.

    Google Scholar 

  22. Cheng S.-J., “Differentiably simple Lie superalgebras and representations of semisimple Lie superalgebras,” J. Algebra, 173, No. 1, 1–43 (1995).

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    A. P. Pozhidaev

  2. University of São Paulo, São Paulo, Brazil

    I. P. Shestakov

  3. Sobolev Institute of Mathematics, Novosibirsk, Russia

    I. P. Shestakov

Authors
  1. A. P. Pozhidaev
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  2. I. P. Shestakov
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Corresponding author

Correspondence to A. P. Pozhidaev.

Additional information

Dedicated to the 70th birthday of Victor Danilovich Mazurov.

Original Russian Text Copyright © 2013 Pozhidaev A.P. and Shestakov I.P.

The first author was supported by the Russian Foundation for Basic Research (Grant 11-01-00938-a) and the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of the Innovative Russia” (Grant 14.740.11.0346). The second author was supported by the FAPESP 2010/ 50347-9 and the CNPq 305344/2009-9.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 2, pp. 389–406, March–April, 2013.

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Pozhidaev, A.P., Shestakov, I.P. Simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0. Sib Math J 54, 301–316 (2013). https://doi.org/10.1134/S0037446613020134

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  • DOI: https://doi.org/10.1134/S0037446613020134

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