Advertisement

Enveloping Algebras of Certain Types of Color Hom-Lie Algebras

  • Abdoreza Armakan
  • Sergei SilvestrovEmail author
Conference paper
  • 33 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)

Abstract

In this paper the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free hom-associative color algebra on a hom-module is described for a certain type of color hom-Lie algebras and is applied to obtain the universal enveloping algebra of those hom-Lie color algebras. Finally, this construction is applied to obtain the extension of the well-known Poincaré–Birkhoff–Witt theorem for Lie algebras to the enveloping algebra of the certain types of color hom-Lie algebra such that some power of the twisting map is the identity map.

Keywords

Color hom-Lie algebra Enveloping algebra 

MSC 2010 Classification

17B75 17B99 17B35 17B37 17D99 16W10 16W55 

Notes

Acknowledgements

Abdoreza Armakan is grateful to the research environment in Mathematics and Applied Mathematics (MAM), Division of Applied Mathematics at the School of Education, Culture and Communication at Mälardalen University, Västerås, Sweden for providing support and excellent research environment during his visits to Mälardalen University when part of the work on this paper has been performed.

References

  1. 1.
    Abdaoui, K., Ammar, F., Makhlouf, A.: Constructions and cohomology of hom-Lie color algebras. Comm. Algebra 43, 4581–4612 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abramov, V., Le Roy, B., Kerner, R.: Hypersymmetry: a \(\mathbb{Z}_3\)-graded generalization of supersymmetry. J. Math. Phys. 38(3), 1650–1669 (1997)MathSciNetzbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Ammar, F., Makhlouf, A.: Hom-Lie superalgebras and hom-Lie admissible superalgebras. J. Algebra 324(7), 1513–1528 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of hom-algebras. J. Lie Theory 21(4), 813–836 (2011). (Zbl 1237.17003, MR2917693)Google Scholar
  6. 6.
    Ammar, F., Mabrouk, S., Makhlouf, A.: Representations and cohomology of \(n\)-ary multiplicative Hom-Nambu-Lie algebras. J. Geom. Phys. (2011).  https://doi.org/10.1016/j.geomphys.2011.04.022
  7. 7.
    Ammar, F., Makhlouf, A., Saadaoui, N.: Chomology of hom-Lie superalgebras and q-deformed Witt superalgebra. Czechoslovak Math. J. 68, 721–761 (2013)zbMATHGoogle Scholar
  8. 8.
    Armakan, A., Silvestrov, S., Farhangdoost, M. R.: Enveloping algebras of hom-Lie color algebras. Turkish J. Math. 43, 316–339 (2019)Google Scholar
  9. 9.
    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 (2014)Google Scholar
  10. 10.
    Arnlind, J., Makhlouf, A., Silvestrov, S.: Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras. J. Math. Phys. 52(12), 123502 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Arnlind, J., Makhlouf, A., Silvestrov, S.: Ternary Hom-Nambu-Lie algebras induced by hom-Lie algebras. J. Math. Phys. 51(4), 043515, 11 pp (2010)Google Scholar
  12. 12.
    Bahturin, Y.A., Mikhalev, A.A., Petrogradsky, V.M., Zaicev, M.V.: Infinite Dimensional Lie Superalgebras. Walter de Gruyter, Berlin (1992)zbMATHGoogle Scholar
  13. 13.
    Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. J. Geom. Phys. 76, 38–60 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Calderon, A., Delgado, J.S.: On the structure of split Lie color algebras. Linear Algebra Appl. 436, 307–315 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cao, Y., Chen, L.: On split regular hom-Lie color algebras. Comm. Algebra 40, 575–592 (2012)MathSciNetGoogle Scholar
  16. 16.
    Casas, J.M., Insua, M.A., Pacheco, N.: On universal central extensions of hom-Lie algebras. Hacet. J. Math. Stat. 44(2), 277–288 (2015). (Zbl 06477457, MR3381108)Google Scholar
  17. 17.
    Chaichian, M., Ellinas, D., Popowicz, Z.: Quantum conformal algebra with central extension. Phys. Lett. B 248, 95–99 (1990)MathSciNetGoogle Scholar
  18. 18.
    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)MathSciNetGoogle Scholar
  19. 19.
    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)MathSciNetGoogle Scholar
  20. 20.
    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)MathSciNetGoogle Scholar
  21. 21.
    Chen, C.W., Petit, T., Van Oystaeyen, F.: Note on cohomology of color Hopf and Lie algebras. J. Algebra 299, 419–442 (2006)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Curtright, T.L., Zachos, C.K.: Deforming maps for quantum algebras. Phys. Lett. B 243, 237–244 (1990)MathSciNetGoogle Scholar
  23. 23.
    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
  24. 24.
    Daskaloyannis, C.: Generalized deformed Virasoro algebras. Modern Phys. Lett. A 7(9), 809–816 (1992)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Guo, L., Zhang, B., Zheng, S.: Universal enveloping algebras and Poincare-Birkhoff-Witt theorem for involutive hom-Lie algebras. J. Lie Theory 28(3), 735–756 (2018). arXiv:1607.05973 [math.QA] (2016)
  26. 26.
    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
  27. 27.
    Hellström, L., Makhlouf, A., Silvestrov, S.D.: Universal algebra applied to hom-associative algebras, and more. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds.), Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics, vol. 85, 157–199. Springer, Berlin (2014)Google Scholar
  28. 28.
    Hellström, L., Silvestrov, S.D.: Commuting Elements in \(q\)-Deformed Heisenberg Algebras, World Scientific, Singapore (2000). ISBN: 981-02-4403-7Google Scholar
  29. 29.
    Hu, N.: \(q\)-Witt algebras, \(q\)-Lie algebras, \(q\)-holomorph structure and representations. Algebra Colloq. 6(1), 51–70 (1999)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kharchenko, V.: Quantum Lie Theory - A Multilinear Approach. Lecture Notes in Mathematics, vol. 2150. Springer International Publishing, Berlin (2015)zbMATHGoogle Scholar
  31. 31.
    Kassel, C.: Cyclic homology of differential operators, the virasoro algebra and a \(q\)-analogue. Comm. Math. Phys. 146(2), 343–356 (1992)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Kerner, R.: Ternary algebraic structures and their applications in physics. In: Proceedings of the BTLP 23rd International Colloquium on Group Theoretical Methods in Physics. arXiv:math-ph/0011023v1 (2000)
  33. 33.
    Kerner, R.: \({\mathbb{Z}}_{3}\)-graded algebras and non-commutative gauge theories, dans le livre. In: Oziewicz, Z., Jancewicz, B., Borowiec, A. (eds.) Spinors, Twistors, Clifford Algebras and Quantum Deformations, 349–357. Kluwer Academic Publishers, Dordrecht (1993)Google Scholar
  34. 34.
    Kerner, R.: The cubic chessboard: geometry and physics. Classical Quantum Gravity 14, A203–A225 (1997)zbMATHGoogle Scholar
  35. 35.
    Kerner, R., Vainerman, L.: On special classes of \(n\)-algebras. J. Math. Phys. 37(5), 2553–2565 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lecomte, P.: On some sequence of graded Lie algebras associated to manifolds. Ann. Global Analysis Geom. 12, 183–192 (1994)MathSciNetzbMATHGoogle Scholar
  37. 37.
    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
  38. 38.
    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 (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
  39. 39.
    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)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Larsson, D., Silvestrov, S.D.: Graded quasi-Lie agebras. Czechoslovak J. Phys. 55, 1473–1478 (2005)MathSciNetGoogle Scholar
  41. 41.
    Larsson, D., Silvestrov, S.D.: Quasi-deformations of \(sl_2(\mathbb{F})\) using twisted derivations. Commun. Algebra 35, 4303–4318 (2007)zbMATHGoogle Scholar
  42. 42.
    Liu, K.Q.: Quantum central extensions. C. R. Math. Rep. Acad. Sci. Canada 13(4), 135–140 (1991)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Liu, K.Q.: Characterizations of the quantum Witt algebra. Lett. Math. Phys. 24(4), 257–265 (1992)Google Scholar
  44. 44.
    Liu, K.Q.: The quantum Witt algebra and quantization of some modules over Witt algebra. Ph.D. Thesis, University of Alberta, Edmonton, Canada, Department of Mathematics (1992)Google Scholar
  45. 45.
    Makhlouf, A.: Paradigm of nonassociative Hom-algebras and Hom-superalgebras. In: Carmona Tapia, J., Morales Campoy, A., Peralta Pereira, A. M., Ramirez Alvarez, M. I. (eds.), Proceedings of Jordan Structures in Algebra and Analysis Meeting. 145–177, Circulo Rojo (2010)Google Scholar
  46. 46.
    Makhlouf, A., Silvestrov, S.D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008). (Zbl 1184.17002, MR2399415). 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
  47. 47.
    Makhlouf, A., Silvestrov, S.D.: Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras (Chapter 17). In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, 189–206. Springer, Berlin (2009). Preprints in Mathematical Sciences, Lund University, Centre for Mathematical Sciences, Centrum Scientiarum Mathematicarum (2007:25) LUTFMA-5091-2007. arXiv:0709.2413 [math.RA] (2007)
  48. 48.
    Makhlouf, A., Silvestrov, S.D.: Hom-algebras and Hom-coalgebras. J. Algebra Appl. 9(4), 553-589 (2010). Preprints in Mathematical Sciences, Lund University, Centre for Mathematical Sciences, Centrum Scientiarum Mathematicarum (2008:19) LUTFMA-5103-2008. arXiv:0811.0400 [math.RA] (2008)
  49. 49.
    Makhlouf, A., Silvestrov, S.: Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22(4), 715–739 (2010). (Zbl 1201.17012, MR2661446). Preprints in Mathematical Sciences, Lund University, Centre for Mathematical Sciences, Centrum Scientiarum Mathematicarum (2007:31) LUTFMA-5095-2007. arXiv:0712.3130v1 [math.RA] (2007)
  50. 50.
    Mikhalev, A.A., Zolotykh, A.A.: Combinatorial Aspects of Lie Superalgebras. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  51. 51.
    Piontkovski, D., Silvestrov, S.D.: Cohomology of 3-dimensional color Lie algebras. J. Algebra 316(2), 499–513 (2007)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Richard, L., Silvestrov, S.D.: Quasi-Lie structure of \(\sigma \)-derivations of \(\mathbb{C}[t^{\pm 1}]\). J. Algebra 319(3), 1285–1304 (2008)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Richard, L., Silvestrov, S.: A note on quasi-Lie and hom-Lie structures of \(\sigma \)-derivations of \(\mathbb{C}[z^{\pm 1}_1,\dots , z^{\pm 1}_n]\). In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.), Generalized Lie Theory in Mathematics, Physics and Beyond, 257–262. Springer, Berlin (2009)Google Scholar
  54. 54.
    Scheunert, M.: Generalized Lie algebras. J. Math. Phys. 20(4), 712–720 (1979)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Scheunert, M.: Graded tensor calculus. J. Math. Phys. 24, 2658–2670 (1983)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Scheunert, M.: Introduction to the cohomology of Lie superalgebras and some applications. Res. Exp. Math. 25, 77–107 (2002)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Scheunert, M., Zhang, R.B.: Cohomology of Lie superalgebras and their generalizations. J. Math. Phys. 39, 5024–5061 (1998)MathSciNetzbMATHGoogle Scholar
  58. 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. 59.
    Sigurdsson, G., Silvestrov, S.: Graded quasi-Lie algebras of Witt type. Czechoslovak J. Phys. 56, 1287–1291 (2006)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Silvestrov, S.: Paradigm of quasi-Lie and quasi-Hom-Lie algebras and quasi-deformations. In: New Techniques in Hopf Algebras and Graded Ring Theory, 165–177. K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels (2007)Google Scholar
  61. 61.
    Sheng, Y.: Representations of hom-Lie algebras. Algebra Represent. Theory 15, 1081–1098 (2012). (Zbl 1294.17001, MR2994017)Google Scholar
  62. 62.
    Sheng, Y., Chen, D.: Hom-Lie 2-algebras. J. Algebra 376, 174–195 (2013). (Zbl 1281.17034, MR3003723)Google Scholar
  63. 63.
    Sheng, Y., Bai, C.: A new approach to hom-Lie bialgebras. J. Algebra 399, 232–250 (2014)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Sheng, Y., Xiong, Z.: On hom-Lie algebras. Linear Multilinear Algebr. 6312, 2379–2395 (2015). (Zbl 06519840, MR3402544)Google Scholar
  65. 65.
    Yau, D.: Enveloping algebra of hom-Lie algebras. J. Gen. Lie Theory Appl. 2(2), 95–108 (2008). (Zbl 1214.17001, MR2399418)Google Scholar
  66. 66.
    Yau, D.: Hom-algebras and homology. J. Lie Theory 19, 409–421 (2009)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Yau, D.: Hom-bialgebras and comodule algebras. Int. Electron. J. Algebra 8, 45–64 (2010)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Yau, D.: Hom-Yang-Baxter equation, hom-Lie algebras, and quasi-triangular bialgebras. J. Phys. A: Math. Theor. 42, 165202 (2009). (Zbl 1179.17001, MR2539278)Google Scholar
  69. 69.
    Yuan, L.: Hom-Lie color algebra structures. Comm. Algebra 40, 575–592 (2012)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Zhou, J., Chen, L., Ma, Y.: Generalized derivations of hom-Lie superalgebras. Acta Math. Sinica (Chin. Ser.) 58, 3737–3751 (2014)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematics and Computer ScienceShahid Bahonar University of KermanKermanIran
  2. 2.Division of Applied Mathematics, School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden

Personalised recommendations