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Algebras with skew-symmetric identity of degree 3

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Abstract

Algebras with one of the following identities are considered:

$$ \begin{array}{*{20}{c}} {\left[ {\left[ {{t_1},\;{t_2}} \right],\;{t_3}} \right] + \left[ {\left[ {{t_2},\;{t_3}} \right],\;{t_1}} \right] + \left[ {\left[ {{t_3},\;{t_1}} \right],\;{t_2}} \right] = 0,} \\ {\left[ {{t_1},\;{t_2}} \right]{t_3} + \left[ {{t_2},\;{t_3}} \right]{t_1} + \left[ {{t_3},\;{t_1}} \right]{t_2} = 0,} \\ {\left\{ {\left[ {{t_1},\;{t_2}} \right],\;{t_3}} \right\} + \left\{ {\left[ {{t_2},\;{t_3}} \right],\;{t_1}} \right\} + \left\{ {\left[ {{t_3},\;{t_1}} \right],\;{t_2}} \right\} = 0,} \\ \end{array} $$

where [t 1 , t 2] = t 1 t 2 − t 2 t 1 and {t 1, t 2} = t 1 t 2 + t 2 t 1 . We prove that any algebra with a skew-symmetric identity of degree 3 is isomorphic or anti-isomorphic to one of such algebras or can be obtained as their q-commutator algebras.

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References

  1. A. A. Balinskii and S. P. Novikov, “Poisson bracket of Hamiltonian type, Frobenius algebras and Lie algebras,” Dokl. AN SSSR, 283, No. 5, 1036–1039 (1985).

    MathSciNet  Google Scholar 

  2. A. S. Dzhumadil’daev, “Novikov–Jordan algebras,” Commun. Algebra, 30, No. 11, 5207–5240 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. S. Dzhumadil’daev and A. Bakirova, “Simple two-sided anti-Lie-admissible algebras,” J. Math. Sci., 161, No. 1, 31–36 (2009).

    Article  Google Scholar 

  4. A. S. Dzhumadil’daev and K. M. Tulenbaev, “Exceptional 0-Alia algebras,” J. Math. Sci., 161, No. 1, 37–40 (2009).

    Article  Google Scholar 

  5. V. Ginsburg and M. M. Kapranov, “Koszul duality for operads,” Duke Math. J., 76, 203–272 (1994).

    Article  MathSciNet  Google Scholar 

  6. J. M. Osborn, “Varieties of algebras,” Adv. Math., 8, 163–369 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  7. J. M. Osborn, “Novikov algebras,” Nova J. Algebra Geom., 1, 1–14 (1992).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Kazakh-British Technical University, Almaty, Kazakhstan

    A. S. Dzhumadil’daev

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  1. A. S. Dzhumadil’daev
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Correspondence to A. S. Dzhumadil’daev.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.

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Dzhumadil’daev, A.S. Algebras with skew-symmetric identity of degree 3. J Math Sci 161, 11–30 (2009). https://doi.org/10.1007/s10958-009-9532-x

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  • DOI: https://doi.org/10.1007/s10958-009-9532-x

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