Advertisement

Algebra and Logic

, Volume 58, Issue 1, pp 77–94 | Cite as

Simple Right-Alternative Unital Superalgebras Over an Algebra of Matrices of Order 2

  • S. V. PchelintsevEmail author
  • O. V. Shashkov
Article

We classify simple right-alternative unital superalgebras over a field of characteristic not 2, whose even part coincides with an algebra of matrices of order 2. It is proved that such a superalgebra either is a Wall double W2|2(ω), or is a Shestakov superalgebra S4|2(σ) (characteristic 3), or is isomorphic to an asymmetric double, an 8-dimensional superalgebra depending on four parameters. In the case of an algebraically closed base field, every such superalgebra is isomorphic to an associative Wall double M2[√1], an alternative 6-dimensional Shestakov superalgebra B4|2 (characteristic 3), or an 8-dimensional Silva–Murakami–Shestakov superalgebra.

Keywords

right-alternative superalgebra simple superalgebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. P. da Silva, L. S. I. Murakami, and I. Shestakov, “On right alternative superalgebras,” Comm. Alg., 44, No. 1, 240-252 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. V. Pchelintsev and O. V. Shashkov, “Simple finite-dimensional right-alternative superalgebras of Abelian type of characteristic zero,” Izv. Ross. Akad. Nauk, Mat., 79, No. 3, 131-158 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. V. Pchelintsev and O. V. Shashkov, “Simple right alternative superalgebras of Abelian type whose even part is a field,” Izv. Ross. Akad. Nauk, Mat., 80, No. 6, 247-257 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. V. Pchelintsev and O. V. Shashkov, “Simple finite-dimensional right alternative superalgebras with unitary even part over a field of characteristic 0,” Mat. Zametki, 100, No. 4, 577-585 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. V. Pchelintsev and O. V. Shashkov, “Simple finite-dimensional right-alternative superalgebras with semisimple strongly associative even part,” Mat. Sb., 208, No. 2, 55-69 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    S. V. Pchelintsev and O. V. Shashkov, “Simple finite-dimensional right-alternative unital superalgebras with strongly associative even part,” Mat. Sb., 208, No. 4, 73-86 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. V. Pchelintsev and O. V. Shashkov, “Simple finite-dimensional right-alternative unital superalgebras with associative-commutative even part over a field of characteristic zero,” Izv. Ross. Akad. Nauk, Mat., 82, No. 3, 136-153 (2018).MathSciNetzbMATHGoogle Scholar
  8. 8.
    N. Svartholm, “On the algebras of relativistic quantum mechanics,” Proc. Roy. Phys. Soc. Lund, 12, 94-108 (1942).MathSciNetzbMATHGoogle Scholar
  9. 9.
    N. Jacobson, Structure and Representations of Jordan Algebras, Colloq. Publ., 39, Am.Math. Soc., Providence, RI (1968).Google Scholar
  10. 10.
    A. M. Slin’ko and I. P. Shestakov, “Right representations of algebras,” Algebra and Logic, 13, No. 5, 312-333 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. I. Murakami and I. Shestakov, “Irreducible unital right alternative bimodules,” J. Alg., 246, No. 2, 897-914 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    S. V. Pchelintsev and O. V. Shashkov, “Simple finite-dimensional right-alternative unital superalgebras over an algebra of matrices of order 2,” Mal’tsev Readings (2017), p. 128.Google Scholar
  13. 13.
    A. A. Albert, “On right alternative algebras,” Ann. Math. (2), 50, 318-328 (1949).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative [in Russian], Nauka, Moscow (1978).zbMATHGoogle Scholar
  15. 15.
    E. Kleinfeld, “Right alternative rings,” Proc. Am. Math. Soc., 4, 939-944 (1953).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Finance Academy under the Government of the Russian FederationMoscowRussia
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Finance Academy under the Government of the Russian FederationMoscowRussia

Personalised recommendations