Skip to main content
SpringerLink
Log in
Book cover

International Conference on Stochastic Processes and Algebraic Structures

SPAS 2017: Algebraic Structures and Applications pp 189–221Cite as

On Classification of Filiform Hom-Lie Algebras

  • Conference paper
  • First Online:

  • 554 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 317))

Abstract

The purpose of this paper is to introduce and study nilpotent and filiform Hom-Lie algebras. Moreover, we extend Vergne and Khakimdjanov’s approach to Hom-type algebras and provide a classification of filiform Hom-Lie algebras of dimension \(n,n\le 7\).

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   229.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Aizawa, N., Sato, H.: \(q\)-deformation of the Virasoro algebra with central extension. Phys. Lett. B 256(1), 185–190 (1991). (Hiroshima University preprint HUPD-9012, 1990)

    Google Scholar 

  2. Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of Hom-algebras. J. Lie Theory 21(4), 813–836 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Ammar, F., Makhlouf, A.: Hom-Lie superalgebras and Hom-Lie admissible superalgebras. J. Algebra 324, 1513–1528 (2010)

    Article  MathSciNet  Google Scholar 

  4. Ancochea-Bermudez, J.M., Goze, M.: Classification des algèbres de Lie filiformes de dimension \(8\). Arch. Math. 50, 511–525 (1988)

    Article  MathSciNet  Google Scholar 

  5. Ancochea-Bermudez, J.M., Goze, M.: Classification des algèbres de Lie nilpotentes de dimension \(7\). Arch. Math. 52(2), 157–185 (1989)

    MATH  Google Scholar 

  6. Ancochea-Bermudez, J.M., Goze, M.: On the varieties of nilpotent Lie algebras of dimension 7 and 8. J. Pure Appl. Algebra 77, 131–140 (1992)

    Article  MathSciNet  Google Scholar 

  7. Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. J. Geom. Phys. 76, 38–60 (2014)

    Article  MathSciNet  Google Scholar 

  8. Boza, L., Echarte, F.J., Ndiiez, J.: Classification of complex filiform Lie algebras of dimension 10. Algebra Groups Geom. 11(3), 253–276 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Bratzlavsky, F.: Classification des algèbres de Lie nilpotentes de dimension \(n\), de classe \(n-1\), dont l’idéal dérivé est commutatif. Acad. Roy. Belg. Bull. Cl. Sci. 560, 858–865 (1974)

    MathSciNet  MATH  Google Scholar 

  10. Gerstenhaber, M.: On the deformation of rings and algebras. Ann. Math. 79, 59–104 (1964)

    Article  MathSciNet  Google Scholar 

  11. Gohr, A.: On Hom-algebras with surjective twisting. J. Algebra 324, 1483–1491 (2010)

    Article  MathSciNet  Google Scholar 

  12. Gomez, J.R., Echarte, F.J.: Classification of complex filiform nilpotent Lie algebras of dimension \(9\). Rend. Sem. Fac. Sci. Univ. Cagliari 61(1), 21–29 (1991)

    MathSciNet  MATH  Google Scholar 

  13. Gomez, J.R., Goze, M., Khakimdjanov, Y.B.: On the \(k\)-abelian filiform Lie algebras. Comm. Algebra 25(2), 431–450 (1997)

    Article  MathSciNet  Google Scholar 

  14. Gomez, J.R., Jimenez-Merchen, A., Khakimdjanov, Y.: Low-dimensional filiform Lie algebras. J. Pure Appl. Algebra 130, 133–158 (1998)

    Article  MathSciNet  Google Scholar 

  15. Goze, M., Khakimdjanov, Y.: Nilpotent Lie algebras. Mathematics and Its Applications, vol. 361. Kluwer Academic Publishers, Boston (1996)

    Google Scholar 

  16. Hartwig, J.T., Larsson, D., Silvestrov, S.D.: Deformations of Lie algebras using \(\sigma \)-derivations. J. Algebra 295, 314–361 (2006)

    Article  MathSciNet  Google Scholar 

  17. Khakimdjanov, Y.: The variety of structures of nilpotent Lie algebras. Uspekhi Mat. Nauk (Russian) 45(6(276)), 151–152 (1990); Translation in Russian Math. Surveys 45(6), 158–159 (1990)

    Google Scholar 

  18. Khakimdjanov, Y.: Variété des lois d’algèbres de Lie nilpotentes. Geom. Dedicata 40, 229–295 (1991)

    Google Scholar 

  19. Larsson, D., Silvestrov, S.D.: Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities. J. Algebra 288, 321–344 (2005)

    Article  MathSciNet  Google Scholar 

  20. Makhlouf, A.: Paradigm of nonassociative Hom-algebras and Hom-superalgebras. In: Proceedings of the “Jordan Structures in Algebra and Analysis” Meeting, Carmona Tapia, J., Morales Campoy, A., Peralta Pereira, A.M., Ramirez Ilvarez, M.I. (eds.) Publishing House: Circulo Rojo, 145–177 (2010)

    Google Scholar 

  21. Makhlouf, A., Silvestrov, S.D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008)

    Article  MathSciNet  Google Scholar 

  22. Makhlouf, A., Silvestrov, S.: Hom-algebras and Hom-coalgebras. J. Algebra Appl. 9(4), 553–589 (2010)

    Article  MathSciNet  Google Scholar 

  23. Makhlouf, A., Silvestrov, S.D.: Notes on formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22(4), 715–759 (2010)

    Article  MathSciNet  Google Scholar 

  24. Morozov, V.: Classification of nilpotent Lie algebras of sixth order. Izv. Vyssh. Uchebn. Zaved. Mat. 4(5), 161–171 (1958)

    MathSciNet  MATH  Google Scholar 

  25. Nijenhuis, A., Richardson, R.W.: Deformation of Lie algebras structures. Jour. Math. Mech. 17, 89–105 (1967)

    MathSciNet  MATH  Google Scholar 

  26. Romdhani, M.: Classification of real and complex nilpotent Lie algebras of dimension \(7\). Linear Multilinear Algebr. 24, 167–189 (1989)

    Article  MathSciNet  Google Scholar 

  27. Seeley, C.: Degenerations of 6-dimensional nilpotent Lie algebras on \(\mathbb{C}\). Comm. Algebra 18(10), 3493–3505 (1990)

    Article  MathSciNet  Google Scholar 

  28. Seeley, C.: Some nilpotent Lie algebras of even dimension. Bull. Austral. Math. Sot. 45, 71–77 (1992)

    Article  MathSciNet  Google Scholar 

  29. Sheng, Y.: Representations of Hom-Lie algebras. Algebr. Represent. Theory 15(6), 1081–1098 (2012)

    Article  MathSciNet  Google Scholar 

  30. Vergne, M.: Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la varieté des algèbres de Lie nilpotentes. Bull. Soc. Math. France 98, 81–116 (1970)

    Google Scholar 

  31. Yau, D.: Enveloping algebra of Hom-Lie algebras. J. Gen. Lie Theory Appl. 2(2), 95–108 (2008)

    Article  MathSciNet  Google Scholar 

  32. Yau, D.: Hom-algebras and homology. J. Lie Theory 19, 409–421 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Yau, D.: Hom-Yang-Baxter equation, Hom-Lie algebras and quasitriangular bialgebras. J. Phys. A 42(16), 165202, 12 pp (2009)

    Google Scholar 

  34. Yau, D.: The Hom-Yang-Baxter equation and Hom-Lie algebras. J. Math. Phys. 52, 053502 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Haute-Alsace, 4 rue des Frères Lumière, 68093, Mulhouse, France

    Abdenacer Makhlouf & Mourad Mehidi

Authors
  1. Abdenacer Makhlouf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mourad Mehidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdenacer Makhlouf .

Editor information

Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Makhlouf, A., Mehidi, M. (2020). On Classification of Filiform Hom-Lie Algebras. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_8

Download citation

Publish with us

Policies and ethics

Search

Navigation