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Jacobi-type identities in algebras and superalgebras

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Abstract

We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of an arbitrary associative algebra. One is a consequence of the other (fundamental identity). From the fundamental identity, we derive a set of four identities (one of which is the Jacobi identity) represented in terms of double commutators and anticommutators. We establish that two of the four identities are independent and show that if the fundamental identity holds for an algebra, then the multiplication operation in that algebra is associative. We find a generalization of the obtained results to the super case and give a generalization of the fundamental identity in the case of arbitrary elements. For nondegenerate even symplectic (super)manifolds, we discuss analogues of the fundamental identity.

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References

  1. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989); English transl. prev. ed. (Grad. Texts Math., Vol. 60), Vol. 60, Springer, Berlin (1978).

    Book  Google Scholar 

  2. N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields [in Russian], Nauka, Moscow (1984); English transl. prev. ed., Wiley-Interscience, New York (1980).

    Google Scholar 

  3. I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B, 69, 309–312 (1977).

    Article  ADS  Google Scholar 

  4. I. A. Batalin and E. S. Fradkin, Phys. Lett. B, 122, 157–164 (1983).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. D. M. Gitman and I. V. Tyutin, Canonical Quantization of Fields with Constraints [in Russian], Nauka, Moscow (1986); English transl.: Quantization of Fields with Constraints, Springer, Berlin (1990).

    MATH  Google Scholar 

  6. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, Princeton (1992).

    MATH  Google Scholar 

  7. D. A. Leites, Russian Math. Surveys, 35, 1–64 (1980).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. B. DeWitt, Supermanifolds, Cambridge Univ. Press, Cambridge (1992).

    Book  MATH  Google Scholar 

  9. I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B, 102, 27–31 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  10. M. Markl and E. Remm, J. Algebra, 299, 171–189 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  11. J.-L. Loday and B. Vallette, Algebraic Operads (Grund. der Math. Wiss., Vol. 346), Springer, Berlin (2012).

    MATH  Google Scholar 

  12. I. A. Batalin and G. A. Vilkovisky, Phys. Rev. D, 28, 2567–2582 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  13. C. Buttin, C. R. Acad. Sci. Paris. Sér. A, 269, 87–89 (1969).

    MATH  MathSciNet  Google Scholar 

  14. F. Wever, Math. Ann., 120, 563–580 (1949).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. D. Blessnohl and H. Laue, Note di Matematika, 8, 111–121 (1988).

    Google Scholar 

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Authors and Affiliations

  1. Tomsk State Pedagogical University, Tomsk, Russia

    P. M. Lavrov & O. V. Radchenko

  2. Tomsk State University, Tomsk, Russia

    P. M. Lavrov & O. V. Radchenko

  3. Lebedev Physical Institute, RAS, Moscow, Russia

    I. V. Tyutin

Authors
  1. P. M. Lavrov
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  2. O. V. Radchenko
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  3. I. V. Tyutin
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Corresponding author

Correspondence to P. M. Lavrov.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 179, No. 2, pp. 196–206, May, 2014.

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Lavrov, P.M., Radchenko, O.V. & Tyutin, I.V. Jacobi-type identities in algebras and superalgebras. Theor Math Phys 179, 550–558 (2014). https://doi.org/10.1007/s11232-014-0161-2

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  • DOI: https://doi.org/10.1007/s11232-014-0161-2

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