Skip to main content
SpringerLink
Log in

On Solvability and Nilpotency for n-Hom-Lie Algebras and \((n+1)\)-Hom-Lie Algebras Induced by n-Hom-Lie Algebras

  • Conference paper
  • First Online:
Algebraic Structures and Applications (SPAS 2017)

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

Abstract

The purpose of this  work is to generalize the concepts of k-solvability and k-nilpotency, initially defined for n-Lie algebras, to n-Hom-Lie algebras and to study their properties. We define k-derived series, k-central descending series and study their properties, we show that k-solvability is a radical property and we apply all of the above to the case of \((n+1)\)-Hom-Lie algebras induced by n-Hom-Lie algebras.

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

Access this chapter

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

Institutional subscriptions

References

  1. Aizawa, N., Sato, H.: \(q\)-deformation of the Virasoro algebra with central extension. Phys. Lett. B 256, 185-190 (1991). (Hiroshima University preprint, 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., Mabrouk, S., Makhlouf, A.: Representation and cohomology of \(n\)-ary multiplicative Hom-Nambu-Lie algebras. J. Geom. Phys. 61, 1898–1913 (2011)

    Article  MathSciNet  Google Scholar 

  4. Arnlind, J., Kitouni, A., Makhlouf, A., Silvestrov, S.: Structure and cohomology of \(3\)-Lie algebras induced by Lie algebras. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds.), Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and & Statistics, vol. 85. Springer, Berlin (2014)

    Google Scholar 

  5. Arnlind, J., Makhlouf, A., Silvestrov, S.: Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras. J. Math. Phys. 51, 043515, 11 pp. (2010)

    Google Scholar 

  6. Arnlind, J., Makhlouf, A., Silvestrov, S.: Construction of \(n\)-Lie algebras and \(n\)-ary Hom-Nambu-Lie algebras. J. Math. Phys. 52, 123502, 13 pp. (2011)

    Google Scholar 

  7. Ataguema, H., Makhlouf, A., Silvestrov, S.: Generalization of n-ary Nambu algebras and beyond. J. Math. Phys. 50, 083501 (2009)

    Google Scholar 

  8. Awata, H., Li, M., Minic, D., Yoneya, T.: On the quantization of Nambu brackets. J. High Energy Phys. 2, Paper 13, 17 pp. (2001)

    Google Scholar 

  9. Bai, R., Bai, C., Wang, J.: Realizations of 3-Lie algebras. J. Math. Phys. 51, 063505 (2010)

    Google Scholar 

  10. Bai, R., Wu, Y., Li, J., Zhou, H.: Constructing \((n+1)\)-Lie algebras from \(n\)-Lie algebras. J. Phys. A 45(47), 475206 (2012)

    Google Scholar 

  11. Bai, R., Song, G., Zhang, Y.: On classification of \(n\)-Lie algebras. Front. Math. China 6, 581–606 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Chaichian, M., Ellinas, D., Popowicz, Z.: Quantum conformal algebra with central extension. Phys. Lett. B 248, 95–99 (1990)

    Article  MathSciNet  Google Scholar 

  14. Chaichian, M., Isaev, A.P., Lukierski, J., Popowic, Z., Prešnajder, P.: \(q\)-deformations of Virasoro algebra and conformal dimensions. Phys. Lett. B 262(1), 32–38 (1991)

    Article  MathSciNet  Google Scholar 

  15. Chaichian, M., Kulish, P., Lukierski, J.: \(q\)-deformed Jacobi identity, \(q\)-oscillators and \(q\)-deformed infinite-dimensional algebras. Phys. Lett. B 237, 401–406 (1990)

    Article  MathSciNet  Google Scholar 

  16. Chaichian, M., Popowicz, Z., Prešnajder, P.: \(q\)-Virasoro algebra and its relation to the \(q\)-deformed KdV system. Phys. Lett. B 249, 63–65 (1990)

    Article  MathSciNet  Google Scholar 

  17. Curtright, T.L., Zachos, C.K.: Deforming maps for quantum algebras. Phys. Lett. B 243, 237–244 (1990)

    Article  MathSciNet  Google Scholar 

  18. Damaskinsky, E.V., Kulish, P.P.: Deformed oscillators and their applications (in Russian). Zap. Nauch. Semin. LOMI 189, 37–74 (1991) (Engl. transl. in J. Sov. Math. 62, 2963–2986 (1992))

    Google Scholar 

  19. Daskaloyannis, C.: Generalized deformed Virasoro algebras. Mod. Phys. Lett. A 7(9), 809–816 (1992)

    Article  MathSciNet  Google Scholar 

  20. Daletskii, Y.L., Takhtajan, L.A.: Leibniz and Lie algebra structures for Nambu algebra. Lett. Math. Phys. 39, 127–141 (1997)

    Article  MathSciNet  Google Scholar 

  21. Filippov, V.T.: \(n\)-Lie algebras. Sib. Math. J. 26, 879–891. Translated from Russian: Sib. Mat. Zh. 26(1985), 126–140 (1985)

    Google Scholar 

  22. Hartwig, J. T., Larsson, D., Silvestrov, S. D.: Deformations of Lie algebras using \(\sigma \)-derivations. J. Algebra 295, 314–361 (2006). (Preprint in Mathematical Sciences 2003:32, LUTFMA-5036-2003, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, 52 pp. (2003))

    Google Scholar 

  23. Hu, N.: \(q\)-Witt algebras, \(q\)-Lie algebras, \(q\)-holomorph structure and representations. Algebra Colloq. 6(1), 51–70 (1999)

    MathSciNet  MATH  Google Scholar 

  24. Kassel, C.: Cyclic homology of differential operators, the virasoro algebra and a \(q\)-analogue. Commun. Math. Phys. 146(2), 343–356 (1992)

    Article  MathSciNet  Google Scholar 

  25. Kasymov, ShM: Theory of \(n\)-Lie algebras. Algebra Logic 26, 155–166 (1987)

    Article  MathSciNet  Google Scholar 

  26. Kitouni, A., Makhlouf, A.: On structure and central extensions of \((n+1)\)-Lie algebras induced by \(n\)-Lie algebras (2014). arXiv:1405.5930

  27. Kitouni, A., Makhlouf, A., Silvestrov, S.: On \((n+1)\)-Hom-Lie algebras induced by \(n\)-Hom-Lie algebras. Georgian Math. J. 23(1), 75–95 (2016)

    Article  MathSciNet  Google Scholar 

  28. Larsson, D., Sigurdsson, G., Silvestrov, S.D.: Quasi-Lie deformations on the algebra \(\mathbb{F}[t]/(t^N)\). J. Gen. Lie Theory Appl. 2, 201–205 (2008)

    Article  MathSciNet  Google Scholar 

  29. Larsson, D., Silvestrov, S. D.: Quasi-Hom-Lie algebras, central extensions and \(2\)-cocycle-like identities. J. Algebra 288, 321–344 (2005). (Preprints in Mathematical Sciences 2004:3, LUTFMA-5038-2004, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University (2004))

    Google Scholar 

  30. Larsson, D., Silvestrov, S.D.: Quasi-Lie algebras. In: Fuchs, J., Mickelsson, J., Rozanblioum, G., Stolin, A., Westerberg, A. (eds.), Noncommutative Geometry and Representation Theory in Mathematical Physics. Contemporary Mathematics, vol. 391, 241–248. American Mathematical Society, Providence, RI (2005). (Preprints in Mathematical Sciences 2004:30, LUTFMA-5049-2004, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University (2004))

    Google Scholar 

  31. Larsson, D., Silvestrov, S.D.: Graded quasi-Lie agebras. Czechoslovak J. Phys. 55, 1473–1478 (2005)

    Article  MathSciNet  Google Scholar 

  32. Larsson, D., Silvestrov, S.D.: Quasi-deformations of \(sl_2(\mathbb{F})\) using twisted derivations. Commun. Algebra 35, 4303–4318 (2007)

    Article  Google Scholar 

  33. Liu, K.Q.: Quantum central extensions. C. R. Math. Rep. Acad. Sci. Can. 13(4), 135–140 (1991)

    MathSciNet  MATH  Google Scholar 

  34. Liu, K.Q.: Characterizations of the quantum Witt algebra. Lett. Math. Phys. 24(4), 257–265 (1992)

    Article  MathSciNet  Google Scholar 

  35. Liu, K.Q.: The Quantum Witt Algebra and Quantization of Some Modules over Witt Algebra. University of Alberta, Edmonton, Canada, Department of Mathematics (1992). Ph.D. Thesis

    Google Scholar 

  36. Makhlouf, A., Silvestrov, S.D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008). (Preprints in Mathematical Sciences 2006:10, LUTFMA-5074-2006, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University (2006))

    Google Scholar 

  37. Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 3(7), 2405–2412 (1973)

    Article  MathSciNet  Google Scholar 

  38. Richard, L., Silvestrov, S.D.: Quasi-Lie structure of \(\sigma \)-derivations of \(\mathbb{C}[t^{\pm 1}]\). J. Algebra 319(3), 1285–1304 (2008)

    Article  MathSciNet  Google Scholar 

  39. Sheng, Y.: Representation of Hom-Lie algebras. Algebr. Reprensent. Theory 15(6), 1081–1098 (2012)

    Article  MathSciNet  Google Scholar 

  40. Sigurdsson, G., Silvestrov, S.: Lie color and Hom-Lie algebras of Witt type and their central extensions. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, 247–255. Springer, Berlin (2009)

    Google Scholar 

  41. Sigurdsson, G., Silvestrov, S.: Graded quasi-Lie algebras of Witt type. Czech. J. Phys. 56, 1287–1291 (2006)

    Article  MathSciNet  Google Scholar 

  42. Takhtajan, L.A.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160(2), 295–315 (1994)

    Article  MathSciNet  Google Scholar 

  43. Takhtajan, L.A.: Higher order analog of Chevalley-Eilenberg complex and deformation theory of \(n\)-gebras. St. Petersburg Math. J. 6(2), 429–438 (1995)

    MathSciNet  Google Scholar 

  44. Yau, D.: A Hom-associative analogue of Hom-Nambu algebras (2010). arXiv: 1005.2373 [math.RA]

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

    Article  MathSciNet  Google Scholar 

  46. Yau, D.: Hom-algebras and homology. J. Lie Theory 19(2), 409–421 (2009)

    MathSciNet  MATH  Google Scholar 

  47. Yau, D.: On \(n\)-ary Hom-Nambu and Hom-Nambu-Lie algebras. J. Geom. Phys. 62, 506–522 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Box 883, 72123, Västerås, Sweden

    Abdennour Kitouni & Sergei Silvestrov

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

    Abdenacer Makhlouf

Authors
  1. Abdennour Kitouni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdenacer Makhlouf
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergei Silvestrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdennour Kitouni .

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

Kitouni, A., Makhlouf, A., Silvestrov, S. (2020). On Solvability and Nilpotency for n-Hom-Lie Algebras and \((n+1)\)-Hom-Lie Algebras Induced by n-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_6

Download citation

Publish with us

Policies and ethics

Search

Navigation