Skip to main content
SpringerLink
Log in

Central extensions of 3-dimensional Zinbiel algebras

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

We describe all central extensions of all 3-dimensional nontrivial complex Zinbiel algebras. As a corollary, we have a full classification of 4-dimensional nontrivial complex Zinbiel algebras and a full classification of 5-dimensional nontrivial complex Zinbiel algebras with 2-dimensional annihilator, which gives the principal step in the algebraic classification of 5-dimensional Zinbiel algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdelwahab, H., Calderón, A.J., Kaygorodov, I.: The algebraic and geometric classification of nilpotent binary Lie algebras. Int. J Algebra Comput. 29(6), 1113–1129 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adashev J., Camacho L., Gomez-Vidal S., Karimjanov I.: Naturally graded Zinbiel algebras with nilindex \(n-3,\) Linear Algebra and its Applications, 443 , 86–104 (2014)

  3. Adashev, J., Camacho, L., Omirov, B.: Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras. J Algebra 479, 461–486 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Adashev, J., Khudoyberdiyev, AKh., Omirov, B.A.: Classifications of some classes of Zinbiel algebras. J Gene. Lie Theory Appl. 4, 10 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Adashev, J., Ladra, M., Omirov, B.: The classification of naturally graded Zinbiel algebras with characteristic sequence equal to \((n-p, p)\). Ukrainian Math. J 71(7), 867–883 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bauerle G.G.A., de Kerf E.A., ten Kroode A.P.E.: Lie Algebras. Part 2. Finite and Infinite Dimensional Lie Algebras and Applications in Physics, edited and with a preface by E.M. de Jager, Studies in Mathematical Physics, vol. 7, North-Holland Publishing Co., Amsterdam, ISBN 0-444-82836-2, 1997, x+554 pp

  7. Bremner, M.: On Tortkara triple systems. Commun. Algebra 46(6), 2396–2404 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calderón, Martín A., Fernández, Ouaridi A., Kaygorodov, I.: The classification of \(n\)-dimensional anticommutative algebras with \((n-3)\)-dimensional annihilator. Communications in Algebra 47(1), 173–181 (2019)

  9. Calderón, Martín A., Fernández, Ouaridi A., Kaygorodov, I.: The classification of 2-dimensional rigid algebras. Linear and Multilinear Algebra 68(4), 828–844 (2020)

  10. Calderón, Martín A., Fernández, Ouaridi A., Kaygorodov, I.: On the classification of bilinear maps with radical of a fixed codimension. Linear and Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1849001

  11. Camacho, L., Cañete, E., Gómez-Vidal, S., Omirov, B.: \(p\)-filiform Zinbiel algebras. Linear Algebra Appl. 438(7), 2958–2972 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Camacho, L., Kaygorodov, I., Lopatkin, V., Salim, M.: The variety of dual Mock-Lie algebras. Commun. Math. 28(2), 161–178 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Camacho, L., Karimjanov, I., Kaygorodov, I., Khudoyberdiev, A.: Central extensions of filiform Zinbiel algebras. Linear and Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1764903

    Article  MATH  Google Scholar 

  14. Cicalò, S., De Graaf, W., Schneider, C.: Six-dimensional nilpotent Lie algebras. Linear Algebra Appl. 436(1), 163–189 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Darijani, I., Usefi, H.: The classification of 5-dimensional \(p\)-nilpotent restricted Lie algebras over perfect fields. I. J Algebra 464, 97–140 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. De Graaf, W.: Classification of nilpotent associative algebras of small dimension. Int. J Algebra Computation 28(1), 133–161 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Graaf, W.: Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. J Algebra 309(2), 640–653 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Diehl, J., Ebrahimi-Fard, K., Tapia, N.: Time warping invariants of multidimensional time series. Acta Applicandae Mathematicae 170, 265–290 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Diehl J., Lyons T., Preis R., Reizenstein J.: Areas of areas generate the shuffle algebra, arXiv:2002.02338

  20. Dokas, I.: Zinbiel algebras and commutative algebras with divided powers. Glasgow Math. J 52(2), 303–313 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dzhumadildaev, A.: Zinbiel algebras under \(q\)-commutators, 0. J Math. Sci. 144(2), 3909–3925 (2007)

    Article  MathSciNet  Google Scholar 

  22. Dzhumadildaev, A., Tulenbaev, K.: Nilpotency of Zinbiel algebras. J Dyn. Control Syst. 11(2), 195–213 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dzhumadildaev, A., Ismailov, N., Mashurov, F.: On the speciality of Tortkara algebras. J Algebra 540, 1–19 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fernández Ouaridi A., Kaygorodov I., Khrypchenko M., Volkov Yu.: Degenerations of nilpotent algebras, arXiv:1905.05361

  25. Gerstenhaber, M.: On the deformation of rings and algebras. Annal Math. 2(79), 59–103 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gorshkov, I., Kaygorodov, I., Khrypchenko, M.: The algebraic classification of nilpotent Tortkara algebras. Commun. Algebra 48(8), 3608–3623 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gorshkov, I., Kaygorodov, I., Khrypchenko, M.: The geometric classification of nilpotent Tortkara algebras. Commun. Algebra 48(1), 204–209 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hegazi, A., Abdelwahab, H.: Classification of five-dimensional nilpotent Jordan algebras. Linear Algebra Appl. 494, 165–218 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hegazi, A., Abdelwahab, H.: Is it possible to find for any \(n, m \in {{\mathbb{N}}}\) a Jordan algebra of nilpotency type \((n,1, m)\)? Beiträge zur Algebra und Geometrie 57(4), 859–880 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hegazi, A., Abdelwahab, H.: The classification of \(n\)-dimensional non-associative Jordan algebras with \((n-3)\)-dimensional annihilator. Commun. Algebra 46(2), 629–643 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hegazi, A., Abdelwahab, H., Calderón, Martín A.: The classification of \(n\)-dimensional non-Lie Malcev algebras with \((n-4)\)-dimensional annihilator. Linear Algebra and its Applications 505, 32–56 (2016)

  32. Hegazi, A., Abdelwahab, H., Calderón, Martín A.: Classification of nilpotent Malcev algebras of small dimensions over arbitrary fields of characteristic not 2. Algebras and Representation Theory 21(1), 19–45 (2018)

  33. Ismailov, N., Kaygorodov, I., Mashurov, F.: The algebraic and geometric classification of nilpotent assosymmetric algebras. Algebras Represent. Theory 24(1), 135–148 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Karimjanov, I., Kaygorodov, I., Khudoyberdiyev, A.: The algebraic and geometric classification of nilpotent Novikov algebras. J Geometry Phy. 143, 11–21 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Karimjanov, I., Kaygorodov, I., Ladra, M.: Central extensions of filiform associative algebras. Linear Multilinear Algebra 69(6), 1083–1101 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kaygorodov I., Khrypchenko M., Lopes S.: The algebraic and geometric classification of nilpotent anticommutative algebras, Journal of Pure and Applied Algebra, 224 , 8, 106337 (2020)

  37. Kaygorodov I., Khrypchenko M., Popov Yu.: The algebraic and geometric classification of nilpotent terminal algebras, Journal of Pure and Applied Algebra, 225 , 6, 106625 (2021)

  38. Kaygorodov, I., Khudoyberdiev, A., Sattarov, A.: One-generated nilpotent terminal algebras. Commun. Algebra 48(10), 4355–4390 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kaygorodov, I., Lopes, S., Páez-Guillán, P.: Non-associative central extensions of null-filiform associative algebras. J Algebra 560, 1190–1210 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kaygorodov, I., Páez-Guillán, P., Voronin, V.: The algebraic and geometric classification of nilpotent bicommutative algebras. Algebras Representation Theory 23(6), 2331–2347 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kaygorodov, I., Popov, Yu., Pozhidaev, A., Volkov, Yu.: Degenerations of Zinbiel and nilpotent Leibniz algebras. Linear Multilinear Algebra 66(4), 704–716 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kaygorodov I., Rakhimov I., Said Husain Sh. K.: The algebraic classification of nilpotent associative commutative algebras, Journal of Algebra and its Applications, 19 , 11, 2050220 (2020)

  43. Loday, J.-L.: Cup-product for Leibniz cohomology and dual Leibniz algebras. Math. Scandinavica 77(2), 189–196 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mukherjee, G., Saha, R.: Cup-product for equivariant Leibniz cohomology and Zinbiel algebras. Algebra Colloquium 26(2), 271–284 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Naurazbekova, A.: On the structure of free dual Leibniz algebras. Eurasian Math. J 10(3), 40–47 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  46. Naurazbekova, A., Umirbaev, U.: Identities of dual Leibniz algebras. TWMS J. Pure Appl. Math. 1(1), 86–91 (2010)

    MathSciNet  MATH  Google Scholar 

  47. Saha, R.: Cup-product in Hom-Leibniz cohomology and Hom-Zinbiel algebras. Commun. Algebra 48(10), 4224–4234 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Skjelbred, T., Sund, T.: Sur la classification des algebres de Lie nilpotentes, C. R. Acad. Sci. Paris Ser. A-B 286(5), A241–A242 (1978)

    MathSciNet  MATH  Google Scholar 

  49. Yau, D.: Deformation of dual Leibniz algebra morphisms. Commun. Algebra 35(4), 1369–1378 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zusmanovich, P.: Central extensions of current algebras. Trans. Am. Math. Soc. 334(1), 143–152 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CMCC, Universidade Federal do ABC, Santo Andre, Brasil

    Ivan Kaygorodov

  2. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Ivan Kaygorodov

  3. Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta, Chile

    María Alejandra Alvarez

  4. Instituto de Ciência e Tecnologia, Universidade Federal de São Paulo, São José dos Campos, Sao Paulo, Brasil

    Thiago Castilho de Mello

Authors
  1. Ivan Kaygorodov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. María Alejandra Alvarez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thiago Castilho de Mello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Kaygorodov.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was started during the research stay of I. Kaygorodov at the Department of Mathematics, parcially funded by the Coloquio de Matemática (CR 4430) of the University of Antofagasta. The work was supported by RFBR 20-01-00030; FAPESP 18/15627-2, 19/03655-4; CNPq 302980/2019-9; AP08052405 of MES RK.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaygorodov, I., Alvarez, M.A. & Mello, T.C.d. Central extensions of 3-dimensional Zinbiel algebras. Ricerche mat 72, 921–947 (2023). https://doi.org/10.1007/s11587-021-00604-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-021-00604-1

Keywords

Mathematics Subject Classification

Search

Navigation