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Multiplicative n-Hom-Lie Color Algebras

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Algebraic Structures and Applications (SPAS 2017)

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

Abstract

The purpose of this paper is to generalize some results on n-Lie algebras and n-Hom-Lie algebras to n-Hom-Lie color algebras. Then we introduce and give some constructions of n-Hom-Lie color algebras.

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References

  1. Abramov, V.: Super 3-Lie algebras induced by super Lie algebras. Adv. Appl. Clifford Algebr. 27(1), 9–16 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

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

    MathSciNet  MATH  Google Scholar 

  4. 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  MATH  Google Scholar 

  5. 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 

  6. 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 

  7. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Bai R., Wang, X., Xiao, W., An, H.: The structure of low dimensional \(n\)-Lie algebras over the field of characteristic \(2\). Linear Algebra Appl. 428(8–9), 1912–1920 (2008)

    Google Scholar 

  14. Bai, R., Chen, L., Meng, D.: The Frattini subalgebra of n-Lie algebras, Acta Math. Sinica, English Series 23(5), 847–856 (2007)

    Google Scholar 

  15. Bai, R., Meng, D.: The central extension of n-Lie algebras. Chin. Ann. Math. 27(4), 491–502 (2006)

    MathSciNet  Google Scholar 

  16. Bai, R., Meng, D.: The centroid of n-Lie algebras. Algebras Groups Geom. 25(2), 29–38 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Bai, R., Zhang, Z., Li, H., Shi, H.: The inner derivation algebras of (n+1)-dimensional n-Lie algebras. Commun. Algebra 28(6), 2927–2934 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bakayoko, I.: Laplacian of Hom-Lie quasi-bialgebras. Int. J. Algebra 8(15), 713–727 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bakayoko, I.: L-modules, L-comodules and Hom-Lie quasi-bialgebras, African Diaspora. J. Math. 17, 49–64 (2014)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Casas, J.M., Loday, J.-L., Pirashvili, T.: Leibniz \(n\)-algebras. Forum Math. 14, 189–207 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. 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 

  24. 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 

  25. 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 

  26. Chen, L., Ma, Y., Ni, L.: Generalized derivations of Lie color algebras. Results Math. 63(3–4), 923–936 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. De Azcárraga, J.A., Izquierdo, J.M.: \(n\)-ary algebras: a review with applications. J. Phys. A: Math. Theor. 43, 293001 (2010)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  33. 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 

  34. Hellström, L., Silvestrov, S.D.: Commuting Elements in \(q\)-Deformed Heisenberg Algebras, World Scientific, Singapore (2000). ISBN: 981-02-4403-7

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Kasymov, Sh.M.: Theory of \(n\)-Lie algebras. Algebra Logic. 26, 155–166 (1987). Translated from Russian: Algebra i Logika 26(3), 277–297 (1987)

    Google Scholar 

  38. Kaygorodov, I.: On \(\delta \)-Derivations of n-ary algebras. Izvestiya: Math. 76(5) 1150–1162 (2012)

    Google Scholar 

  39. Kaygorodov, I.: \((n + 1)\)-Ary derivations of simple n-ary algebras. Algebra Logic 50(5), 470–471 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kaygorodov, I.: \((n + 1)\)-Ary derivations of semisimple Filippov algebras. Math. Notes 96(2), 208–216 (2014)

    Article  MathSciNet  Google Scholar 

  41. Kaygorodov, I., Popov, Y.: Generalized derivations of (color) \(n\)-ary algebras. Linear Multilinear Algebra 64(6), 1086–1106 (2016)

    Google Scholar 

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

  43. 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)

    MathSciNet  MATH  Google Scholar 

  44. 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  MATH  Google Scholar 

  45. 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 

  46. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  49. Ling, W.X.: On the structure of \(n\)-Lie algebras. University-GHS-Siegen, Siegen (1993). Ph.D. Thesis

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. 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 

  53. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  57. Rotkiewicz, M.: Cohomology ring of n-Lie algebras. Extracta Math. 20, 219–232 (2005)

    MathSciNet  MATH  Google Scholar 

  58. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  61. 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 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Ibrahima Bakayoko is grateful to the research environment in Mathematics and Applied Mathematics MAM, the Division of Applied Mathematics of the School of Education, Culture and Communication at Mälardalen University for hospitality and an excellent and inspiring environment for research and research education and cooperation in Mathematics during his visit, in the framework of research and research education capacity and cooperation development in Africa and programs in Mathematics between Sweden and countries in Africa supported by Swedish International Development Agency (Sida) and International Program in Mathematical Sciences (IPMS).

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Authors and Affiliations

  1. Département de Mathématiques, Université de N’Zérékoré, BP 50, Nzerekore, Guinea

    Ibrahima Bakayoko

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

    Sergei Silvestrov

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  1. Ibrahima Bakayoko
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  2. Sergei Silvestrov
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Correspondence to Ibrahima Bakayoko .

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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

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Bakayoko, I., Silvestrov, S. (2020). Multiplicative n-Hom-Lie Color 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_7

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