Skip to main content

The factorization problem for Jordan algebras: applications

Abstract

We investigate the factorization problem as well as the classifying complements problem in the setting of Jordan algebras. Matched pairs of Jordan algebras and the corresponding bicrossed products are introduced. It is shown that any Jordan algebra which factorizes through two given Jordan algebras is isomorphic to a bicrossed product associated to a certain matched pair between the same two Jordan algebras. Furthermore, a new type of deformation of a Jordan algebra is proposed as the main step towards solving the classifying complements problem.

This is a preview of subscription content, access via your institution.

References

  1. Agore, A.L.: Classifying complements for associative algebras. Linear Algebra Appl. 446, 345–355 (2014)

    MathSciNet  Article  Google Scholar 

  2. Agore, A.L.: Classifying bicrossed products of two Taft algebras. J. Pure Appl. Algebra 222, 914–930 (2018)

    MathSciNet  Article  Google Scholar 

  3. Agore, A.L., Bontea, C.G., Militaru, G.: Classifying bicrossed products of Hopf algebras. Algebr. Represent. Theory 17, 227–264 (2014)

    MathSciNet  Article  Google Scholar 

  4. Agore, A.L., Chirvăsitu, A., Ion, B., Militaru, G.: Bicrossed products for finite groups. Algebr. Represent. Theory 12, 481–488 (2009)

    MathSciNet  Article  Google Scholar 

  5. Agore, A.L., Militaru, G.: Classifying complements for Hopf algebras and Lie algebras. J. Algebra 391, 193–208 (2013)

    MathSciNet  Article  Google Scholar 

  6. Agore, A.L., Militaru, G.: Unified products for Leibniz algebras. Applications. Linear Algebra Appl. 439, 2609–2633 (2013)

    MathSciNet  Article  Google Scholar 

  7. Agore, A..L., Militaru, G.: Classifying complements for groups. Ann. Inst. Fourier Appl. 65, 1349–1365 (2015)

    MathSciNet  Article  Google Scholar 

  8. Agore, A.L., Năstăsescu, L.: Bicrossed products with the Taft algebra. Arch. Math. 113, 21–36 (2019)

    MathSciNet  Article  Google Scholar 

  9. Agore, A.L., Militaru, G.: Unified products for Jordan algebras. Applications, arXiv:2107.04970, submitted for publication

  10. Ancochea Bermudez, J.M., Campoamor-Stursberg, R., Garcia Vergnolle, L., Sanchez Hernandez, J.: Contractions de algebres de Jordan en dimension 2. J. Algebra 319, 2395–2409 (2008)

    MathSciNet  Article  Google Scholar 

  11. Brzeziński, T.: Deformation of algebra factorisations. Commun. Algebra 29, 737–748 (2001)

    MathSciNet  Article  Google Scholar 

  12. Caenepeel, S., Ion, B., Militaru, G., Zhu, S.: The factorization problem and the smash biproduct of algebras and coalgebras. Algebr. Represent. Theory 3, 19–42 (2000)

    MathSciNet  Article  Google Scholar 

  13. Cap, A., Schichl, H., Vanzura, J.: On twisted tensor product of algebras. Commun. Algebra 23, 4701–4735 (1995)

    MathSciNet  Article  Google Scholar 

  14. Carotenuto, A., Dabrowski, L., Dubois-Violette, M.: Differential calculus on Jordan algebra and Jordan modules. Lett. Math. Phys. 109, 113–133 (2019)

    MathSciNet  Article  Google Scholar 

  15. Gelaki, S.: Exact factorizations and extensions of fusion categories. J. Algebra 480, 505–518 (2017)

    MathSciNet  Article  Google Scholar 

  16. Jacobson, N.: Structure and representations of Jordan Algebras. Am. Math. Soc. (1968)

  17. Kashuba, I.: Variety of Jordan algebras in small dimension. Algebra Discrete Math. 2, 62–76 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Kashuba, I., Ovsienko, O., Shestakov, I.: Representation type of Jordan algebras. Adv. Math. 226, 385–418 (2011)

    MathSciNet  Article  Google Scholar 

  19. Keilberg, M.: Automorphisms of the doubles of purely non-abelian finite groups. Algebr. Represent. Theory 18, 1267–1297 (2015)

    MathSciNet  Article  Google Scholar 

  20. Koecher, M.: The Minnesota Notes on Jordan Algebras and their Applications. Lecture Notes in Math, vol. 1710. Springer, Berlin (1999)

  21. Kosmann-Schwarzbach, Y., Magri, F.: Poisson-Lie groups and complete integrability. I. Drinfel’d bialgebras, dual extensions and their canonical representations. Ann. Inst. H. Poincare Phys. Theor 49, 433–460 (1988)

    MathSciNet  MATH  Google Scholar 

  22. Lu, J.H., Weinstein, A.: Poisson Lie groups, dressing transformations and Bruhat decompositions. J. Differ. Geom. 31, 501–526 (1990)

    MathSciNet  Article  Google Scholar 

  23. Maillet, E.: Sur les groupes échangeables et les groupes décomposables. Bull. Soc. Math. France 28, 7–16 (1900)

    MathSciNet  Article  Google Scholar 

  24. Majid, S.: Matched pairs of Lie groups and Hopf algebra bicrossproducts. Nuclear Phys. B 6, 422–424 (1989)

    MathSciNet  Article  Google Scholar 

  25. Majid, S.: Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130, 17–64 (1990)

    MathSciNet  Article  Google Scholar 

  26. McCrimmon, K.: A Taste of Jordan Algebras. Universitext, Springer (2004)

    MATH  Google Scholar 

  27. Ore, O.: Structures and group theory. I. Duke Math. J. 3(2), 149–174 (1937)

    MathSciNet  Article  Google Scholar 

  28. Takeuchi, M.: Matched pairs of groups and bismash products of Hopf algebras. Commun. Algebra 9, 841–882 (1981)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, project number PN-III-P4-ID-PCE-2020-0458, within PNCDI III.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. L. Agore.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Agore, A.L., Militaru, G. The factorization problem for Jordan algebras: applications. Collect. Math. (2022). https://doi.org/10.1007/s13348-022-00369-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13348-022-00369-2

Keywords

  • Matched pair of Jordan algebras
  • Bicrossed products of Jordan algebras
  • The factorization problem for Jordan algebras
  • Deformations of a Jordan algebra

Mathematics Subject Classification

  • 17C10
  • 17C50
  • 17C55