Skip to main content
Top

2020 | OriginalPaper | Chapter

Quadratic Color Hom-Lie Algebras

Authors : Faouzi Ammar, Imen Ayadi, Sami Mabrouk, Abdenacer Makhlouf

Published in: Associative and Non-Associative Algebras and Applications

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The purpose of this paper is to study quadratic color Hom-Lie algebras. We present some constructions of quadratic color Hom-Lie algebras which we use to provide several examples. We describe \(T^*\)-extensions and central extensions of color Hom-Lie algebras and establish some cohomological characterizations.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Aizawa, N., Sato, H.: q-deformation of the Virasoro algebra with central extension. Phys. Lett. B 256, 185–190 (1991)MathSciNetCrossRef Aizawa, N., Sato, H.: q-deformation of the Virasoro algebra with central extension. Phys. Lett. B 256, 185–190 (1991)MathSciNetCrossRef
2.
3.
go back to reference Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of Hom-algebras. J. Lie Theory 21(4), 813–836 (2011)MathSciNetMATH Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of Hom-algebras. J. Lie Theory 21(4), 813–836 (2011)MathSciNetMATH
4.
go back to reference Ammar, F., Mabrouk, S., Makhlouf, A.: Constructions of quadratic \(n\)-ary Hom-Nambu algebras. In: Algebra, Geometry and Mathematical Physics, Springer Proceedings in Mathematics & Statistics, vol. 85, pp. 193–224 (2014) Ammar, F., Mabrouk, S., Makhlouf, A.: Constructions of quadratic \(n\)-ary Hom-Nambu algebras. In: Algebra, Geometry and Mathematical Physics, Springer Proceedings in Mathematics & Statistics, vol. 85, pp. 193–224 (2014)
5.
go back to reference Ammar, F., Makhlouf, A.: Hom-Lie superalgebras and Hom-Lie admissible superalgebras. J. Algebr. 324, 1513–1528 (2010)MathSciNetCrossRef Ammar, F., Makhlouf, A.: Hom-Lie superalgebras and Hom-Lie admissible superalgebras. J. Algebr. 324, 1513–1528 (2010)MathSciNetCrossRef
6.
go back to reference Ayadi, I., Benamor, H., Benayadi, S.: Lie superalgebras with some homogeneous structures. J. Algebr. Appl. 11(05), 1250095 (2012)MathSciNetCrossRef Ayadi, I., Benamor, H., Benayadi, S.: Lie superalgebras with some homogeneous structures. J. Algebr. Appl. 11(05), 1250095 (2012)MathSciNetCrossRef
7.
go back to reference Bajo, I., Benayadi, S., Bordemann, M.: Generalized double extension and descriptions of quadratic Lie superalgebras (2007). arXiv:0712.0228 Bajo, I., Benayadi, S., Bordemann, M.: Generalized double extension and descriptions of quadratic Lie superalgebras (2007). arXiv:​0712.​0228
8.
9.
go back to reference Benayadi, S.: Quadratic Lie superalgebras with the completely reducible action of the even part on the odd part. J. Algebr. 223, 344–366 (2000)MathSciNetCrossRef Benayadi, S.: Quadratic Lie superalgebras with the completely reducible action of the even part on the odd part. J. Algebr. 223, 344–366 (2000)MathSciNetCrossRef
10.
11.
go back to reference Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. J. Geom. Phys. 76, 38–60 (2014)MathSciNetCrossRef Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. J. Geom. Phys. 76, 38–60 (2014)MathSciNetCrossRef
12.
go back to reference Bordemann, M.: NonDegenerate invariant bilinear forms in nonassociative algebras. Acta Math. Univ. Comenian LXVI(2), 151–201 (1997) Bordemann, M.: NonDegenerate invariant bilinear forms in nonassociative algebras. Acta Math. Univ. Comenian LXVI(2), 151–201 (1997)
13.
go back to reference Chaichian, M., Ellinas, D., Popowicz, Z.: Quantum conformal algebra with central extension. Phys. Lett. B 248, 95–99 (1990)MathSciNetCrossRef Chaichian, M., Ellinas, D., Popowicz, Z.: Quantum conformal algebra with central extension. Phys. Lett. B 248, 95–99 (1990)MathSciNetCrossRef
14.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
15.
go back to reference De Medeiros, P., Figueroa-O’Farrill, J., Méndez-Escobar, E., Ritter, P.: On the Lie-algebraic origin of metric 3-algebras. Commun. Math. Phys. 290(3), 871–902 (2009)MathSciNetCrossRef De Medeiros, P., Figueroa-O’Farrill, J., Méndez-Escobar, E., Ritter, P.: On the Lie-algebraic origin of metric 3-algebras. Commun. Math. Phys. 290(3), 871–902 (2009)MathSciNetCrossRef
16.
go back to reference Elhamdadi, M., Makhlouf, A.: Deformations of Hom-Alternative and Hom-Malcev algebras. Algebr. Groups Geom. 28(2), 117–145 (2011)MathSciNetMATH Elhamdadi, M., Makhlouf, A.: Deformations of Hom-Alternative and Hom-Malcev algebras. Algebr. Groups Geom. 28(2), 117–145 (2011)MathSciNetMATH
19.
go back to reference Hartwig, J., Larsson, D., Silvestrov, S.: Deformations of Lie algebras using \(\sigma \) -derivations. J. Algebr. 295, 314–361 (2006)MathSciNetCrossRef Hartwig, J., Larsson, D., Silvestrov, S.: Deformations of Lie algebras using \(\sigma \) -derivations. J. Algebr. 295, 314–361 (2006)MathSciNetCrossRef
20.
go back to reference Hegazi, A.: Classification of nilpotent Lie superalgebras of dimension five. I. Int. J. Theor. Phys. 38(6), 1735–1739 (1999)MathSciNetCrossRef Hegazi, A.: Classification of nilpotent Lie superalgebras of dimension five. I. Int. J. Theor. Phys. 38(6), 1735–1739 (1999)MathSciNetCrossRef
21.
go back to reference Larsson, D., Silvestrov, S.: Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities. J. Algebr. 288, 321–344 (2005)MathSciNetCrossRef Larsson, D., Silvestrov, S.: Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities. J. Algebr. 288, 321–344 (2005)MathSciNetCrossRef
22.
go back to reference Makhlouf, A., Silvestrov, S.: Notes on formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22(4), 715–759 (2010)MathSciNetCrossRef Makhlouf, A., Silvestrov, S.: Notes on formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22(4), 715–759 (2010)MathSciNetCrossRef
23.
go back to reference Makhlouf, A.: Hom-alternative algebras and Hom-Jordan algebras. Int. Electron. J. Algebr. 8, 177–190 (2010)MathSciNetMATH Makhlouf, A.: Hom-alternative algebras and Hom-Jordan algebras. Int. Electron. J. Algebr. 8, 177–190 (2010)MathSciNetMATH
25.
go back to reference Medina, A., Revoy, P.: Algèbres de Lie et produit scalaire invariant. Ann. Sci. Ecole Norm. Sup. (4) 18, 553–561 (1985)MathSciNetCrossRef Medina, A., Revoy, P.: Algèbres de Lie et produit scalaire invariant. Ann. Sci. Ecole Norm. Sup. (4) 18, 553–561 (1985)MathSciNetCrossRef
26.
go back to reference Qingcheng, Z., Yongzheng, Z.: Derivations and extensions of Lie color algebra. Acta Math. Sci. 28, 933–948 (2008)MathSciNetCrossRef Qingcheng, Z., Yongzheng, Z.: Derivations and extensions of Lie color algebra. Acta Math. Sci. 28, 933–948 (2008)MathSciNetCrossRef
28.
go back to reference Rittenberg, V., Wyler, D.: Sequences of graded \(\mathbb{Z}\otimes \mathbb{Z}\) Lie algebras and superalgebras. J. Math. Phys. 19, 2193 (1978)MathSciNetCrossRef Rittenberg, V., Wyler, D.: Sequences of graded \(\mathbb{Z}\otimes \mathbb{Z}\) Lie algebras and superalgebras. J. Math. Phys. 19, 2193 (1978)MathSciNetCrossRef
29.
go back to reference Scheunert, M.: The theory of Lie Superalgebras: An Introduction. Springer (1979) Scheunert, M.: The theory of Lie Superalgebras: An Introduction. Springer (1979)
30.
go back to reference Scheunert, M., Zhang, R.B.: Cohomology of Lie superalgebras and their generalizations. J. Math. Phys. 39, 5024 (1998)MathSciNetCrossRef Scheunert, M., Zhang, R.B.: Cohomology of Lie superalgebras and their generalizations. J. Math. Phys. 39, 5024 (1998)MathSciNetCrossRef
31.
go back to reference Scheunert, M.: Generalized Lie algebras. In: Group Theoretical Methods in Physics, vol. 450–450 (1979) Scheunert, M.: Generalized Lie algebras. In: Group Theoretical Methods in Physics, vol. 450–450 (1979)
34.
go back to reference Wang, S., Zhu, L., Su, Y.: Non-degenerate invariant bilinear forms on Lie color algebras. Algebr. Colloq. 17, 365–374 (2010)MathSciNetCrossRef Wang, S., Zhu, L., Su, Y.: Non-degenerate invariant bilinear forms on Lie color algebras. Algebr. Colloq. 17, 365–374 (2010)MathSciNetCrossRef
37.
go back to reference Yau, D.: Hom-Maltsev, hom-alternative and hom-Jordan algebras. Int. Electron. J. Algebr. 11, 177–217 (2012)MathSciNetMATH Yau, D.: Hom-Maltsev, hom-alternative and hom-Jordan algebras. Int. Electron. J. Algebr. 11, 177–217 (2012)MathSciNetMATH
39.
go back to reference Zhang, R., Hou, D., Bai, C.: A Hom-version of the affinizations of Balinskii-Novikov and Novikov superalgebras. J. Math. Phys. 52, 023505 (2011)MathSciNetCrossRef Zhang, R., Hou, D., Bai, C.: A Hom-version of the affinizations of Balinskii-Novikov and Novikov superalgebras. J. Math. Phys. 52, 023505 (2011)MathSciNetCrossRef
Metadata
Title
Quadratic Color Hom-Lie Algebras
Authors
Faouzi Ammar
Imen Ayadi
Sami Mabrouk
Abdenacer Makhlouf
Copyright Year
2020
DOI
https://doi.org/10.1007/978-3-030-35256-1_16

Premium Partner