Universal central extension of direct limits of Hom-Lie algebras

Valiollah Khalili

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 1, page 275-293
  • ISSN: 0011-4642

Abstract

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We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras ( ℒ i , α ℒ i ) is (isomorphic to) the direct limit of universal central extensions of ( ℒ i , α ℒ i ) . As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras { ( sl k ( å ) , α k ) } k ∈ I and describe the universal central extension of its direct limit.

How to cite

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Khalili, Valiollah. "Universal central extension of direct limits of Hom-Lie algebras." Czechoslovak Mathematical Journal 69.1 (2019): 275-293. <http://eudml.org/doc/294610>.

@article{Khalili2019,
abstract = {We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $(\mathcal \{L\}_i, \alpha _\{\mathcal \{L\}_i\})$ is (isomorphic to) the direct limit of universal central extensions of $(\mathcal \{L\}_i, \alpha _\{\mathcal \{L\}_i\})$. As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras $\lbrace (\{\rm sl\}_\{k\}(å), \alpha _k)\rbrace _\{k\in I\}$ and describe the universal central extension of its direct limit.},
author = {Khalili, Valiollah},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hom-Lie algebra; extension of Hom-Lie algebras and its direct limit},
language = {eng},
number = {1},
pages = {275-293},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Universal central extension of direct limits of Hom-Lie algebras},
url = {http://eudml.org/doc/294610},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Khalili, Valiollah
TI - Universal central extension of direct limits of Hom-Lie algebras
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 275
EP - 293
AB - We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $(\mathcal {L}_i, \alpha _{\mathcal {L}_i})$ is (isomorphic to) the direct limit of universal central extensions of $(\mathcal {L}_i, \alpha _{\mathcal {L}_i})$. As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras $\lbrace ({\rm sl}_{k}(å), \alpha _k)\rbrace _{k\in I}$ and describe the universal central extension of its direct limit.
LA - eng
KW - Hom-Lie algebra; extension of Hom-Lie algebras and its direct limit
UR - http://eudml.org/doc/294610
ER -

References

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  1. Alison, B., Benkart, G., Gao, Y., 10.1007/s002080050341, Math. Ann. 316 (2000), 499-527. (2000) Zbl0989.17004MR1752782DOI10.1007/s002080050341
  2. Ammar, F., Ejbehi, Z., Makhlouf, A., Cohomology and deformations of Hom-algebras, J. Lie Theory 24 (2011), 813-836. (2011) Zbl1237.17003MR2917693
  3. Ammar, F., Mobrouk, S., Makhlouf, A., 10.1016/j.geomphys.2011.04.022, J. Geom. Phys. 61 (2011), 1898-1913. (2011) Zbl1258.17007MR2822457DOI10.1016/j.geomphys.2011.04.022
  4. Periñán, M. J. Aragón, Martín, A. J. Calderón, 10.13001/1081-3810.1579, Electron. J. Linear Algebra 24 (2012), 45-65. (2012) Zbl1258.16045MR2994641DOI10.13001/1081-3810.1579
  5. Arnal, D., Bakbrahem, W., Makhlouf, A., Quadratic and Pinczon algebras, Available at https://arxiv.org/pdf/1603.00435. 
  6. Azam, S., Behbodi, G., Yousofzadeh, M., 10.1080/00927872.2016.1172600, Commun. Algebra 44 (2016), 5309-5341. (2016) Zbl06638003MR3520278DOI10.1080/00927872.2016.1172600
  7. Benayadi, S., Makhlouf, A., 10.1016/j.geomphys.2013.10.010, J. Geom. Phys. 76 (2014), 38-60. (2014) Zbl1331.17028MR3144357DOI10.1016/j.geomphys.2013.10.010
  8. Bloch, S., 10.1007/bfb0089515, Algebra -Theory, 1980 Lecture Notes in Math. 854 Springer, Berlin (1981), 1-23. (1981) Zbl0469.14009MR0618298DOI10.1007/bfb0089515
  9. Bourbaki, N., 10.1007/978-3-540-33850-5, Hermann, Paris (1970), French. (1970) Zbl0211.02401MR0274237DOI10.1007/978-3-540-33850-5
  10. Casas, J. M., Insua, M. A., Pacheco, N., 10.15672/hjms.2015449110, Hacet. J. Math Stat. 44 (2015), 277-288. (2015) Zbl1344.17003MR3381108DOI10.15672/hjms.2015449110
  11. Cheng, Y., Su, Y., 10.1007/s10114-011-9626-5, Acta Math. Sin., Engl. Ser. 27 (2011), 813-830. (2011) Zbl1250.17001MR2786445DOI10.1007/s10114-011-9626-5
  12. Cheng, Y., Su, Y., 10.1142/S1005386713000266, Algebra Colloq. 20 (2013), 299-308. (2013) Zbl1310.17007MR3043314DOI10.1142/S1005386713000266
  13. Frégier, Y., Gohr, A., Silvestrov, S. D., 10.4303/jglta/S090402, J. Gen. Lie Theory Appl. 3 (2009), 285-295. (2009) Zbl1237.17005MR2602991DOI10.4303/jglta/S090402
  14. Gao, Y., Shang, S., 10.1016/j.jalgebra.2006.10.044, J. Algebra. 311 (2007), 216-230. (2007) Zbl1136.17016MR2309885DOI10.1016/j.jalgebra.2006.10.044
  15. Hartwig, J. T., Larsson, D., Silvestrov, S. D., 10.1016/j.jalgebra.2005.07.036, J. Algebra 295 (2006), 314-361. (2006) Zbl1138.17012MR2194957DOI10.1016/j.jalgebra.2005.07.036
  16. Jin, Q., Li, X., 10.1016/j.jalgebra.2007.12.005, J. Algebra 319 (2008), 1398-1408. (2008) Zbl1144.17005MR2383052DOI10.1016/j.jalgebra.2007.12.005
  17. Kassel, C., Loday, J.-L., 10.5802/aif.896, Ann. Inst. Fourier 32 (1982), 119-142 French. (1982) Zbl0485.17006MR0694130DOI10.5802/aif.896
  18. Khalili, V., 10.4134/BKMS.2012.49.6.1199, Bull. Korean Math. Soc. 49 (2012), 1199-1211. (2012) Zbl1276.17013MR3002679DOI10.4134/BKMS.2012.49.6.1199
  19. Li, X., 10.1155/2013/938901, Adv. Math. Phys. 2013 Article ID 938901, 7 pages. Zbl1291.17010MR3132688DOI10.1155/2013/938901
  20. Makhlouf, A., Silvestrov, S., 10.4303/jglta/S070206, J. Gen. Lie Theory Appl. 2 (2008), 51-64. (2008) Zbl1184.17002MR2399415DOI10.4303/jglta/S070206
  21. Makhlouf, A., Silvestrov, S., 10.1515/FORUM.2010.040, Forum Math. 22 (2010), 715-739. (2010) Zbl1201.17012MR2661446DOI10.1515/FORUM.2010.040
  22. Moody, R. V., Pianzola, A., Lie Algebras with Triangular Decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York (1995). (1995) Zbl0874.17026MR1323858
  23. Neeb, K.-H., 10.1023/A:1019743224737, Acts Appl. Math. 73 (2002), 175-219. (2002) Zbl1019.22011MR1926500DOI10.1023/A:1019743224737
  24. Neher, E., 10.1007/978-1-4613-0235-3_10, Groups, Rings, Lie and Hopf Algebras, 2001 Math. Appl. 555, Kluwer Academic Publishers, Dordrecht Y. Bahturin (2003), 141-166. (2003) Zbl1077.17016MR1995057DOI10.1007/978-1-4613-0235-3_10
  25. Neher, E., Sun, J., 10.1016/j.jalgebra.2012.06.020, J. Algebra 368 (2012), 169-181. (2012) Zbl1301.17007MR2955226DOI10.1016/j.jalgebra.2012.06.020
  26. Sheng, Y., 10.1007/s10468-011-9280-8, Alger. Represent. Theory 15 (2012), 1081-1098. (2012) Zbl1294.17001MR2994017DOI10.1007/s10468-011-9280-8
  27. Kallen, W. L. J. van der, 10.1007/BFb0060175, Lecture Notes in Mathematics 356, Springer, Berlin (1973). (1973) Zbl0275.17006MR0364484DOI10.1007/BFb0060175
  28. Weibel, C. A., 10.1017/CBO9781139644136, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge (1994). (1994) Zbl0797.18001MR1269324DOI10.1017/CBO9781139644136
  29. Yau, D., 10.4303/jglta/S070209, J. Gen. Lie Theory Appl. 2 (2008), 95-108. (2008) Zbl1214.17001MR2399418DOI10.4303/jglta/S070209
  30. Yau, D., Hom-algebras and homology, J. Lie Theory 19 (2009), 409-421. (2009) Zbl1252.17002MR2572137

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