| Title:
|
Universal central extension of direct limits of Hom-Lie algebras (English) |
| Author:
|
Khalili, Valiollah |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
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0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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69 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
275-293 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\{({\rm sl}_{k}(å), \alpha _k)\}_{k\in I}$ and describe the universal central extension of its direct limit. (English) |
| Keyword:
|
Hom-Lie algebra |
| Keyword:
|
extension of Hom-Lie algebras and its direct limit |
| MSC:
|
17A30 |
| MSC:
|
17B55 |
| MSC:
|
17B60 |
| MSC:
|
17B99 |
| idZBL:
|
Zbl 07088784 |
| idMR:
|
MR3923589 |
| DOI:
|
10.21136/CMJ.2018.0290-17 |
| . |
| Date available:
|
2019-03-08T15:02:11Z |
| Last updated:
|
2021-04-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147632 |
| . |
| Reference:
|
[1] Alison, B., Benkart, G., Gao, Y.: Central extensions of Lie algebras graded by finite root systems.Math. Ann. 316 (2000), 499-527. Zbl 0989.17004, MR 1752782, 10.1007/s002080050341 |
| Reference:
|
[2] Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of Hom-algebras.J. Lie Theory 24 (2011), 813-836. Zbl 1237.17003, MR 2917693 |
| Reference:
|
[3] Ammar, F., Mobrouk, S., Makhlouf, A.: Representations and cohomology of $n$-ary multiplicative Hom-Nambu-Lie algebras.J. Geom. Phys. 61 (2011), 1898-1913. Zbl 1258.17007, MR 2822457, 10.1016/j.geomphys.2011.04.022 |
| Reference:
|
[4] Periñán, M. J. Aragón, Martín, A. J. Calderón: On graded matrix Hom-algebras.Electron. J. Linear Algebra 24 (2012), 45-65. Zbl 1258.16045, MR 2994641, 10.13001/1081-3810.1579 |
| Reference:
|
[5] Arnal, D., Bakbrahem, W., Makhlouf, A.: Quadratic and Pinczon algebras.Available at https://arxiv.org/pdf/1603.00435. |
| Reference:
|
[6] Azam, S., Behbodi, G., Yousofzadeh, M.: Direct union of Lie tori (realization of locally extended affine Lie algebras).Commun. Algebra 44 (2016), 5309-5341. Zbl 06638003, MR 3520278, 10.1080/00927872.2016.1172600 |
| Reference:
|
[7] Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms.J. Geom. Phys. 76 (2014), 38-60. Zbl 1331.17028, MR 3144357, 10.1016/j.geomphys.2013.10.010 |
| Reference:
|
[8] Bloch, S.: The dilogarithm and extensions of Lie algebras.Algebra $K$-Theory, 1980 Lecture Notes in Math. 854 Springer, Berlin (1981), 1-23. Zbl 0469.14009, MR 0618298, 10.1007/bfb0089515 |
| Reference:
|
[9] Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitres 1 à 3.Hermann, Paris (1970), French. Zbl 0211.02401, MR 0274237, 10.1007/978-3-540-33850-5 |
| Reference:
|
[10] Casas, J. M., Insua, M. A., Pacheco, N.: On universal central extensions of Hom-Lie algebras.Hacet. J. Math Stat. 44 (2015), 277-288. Zbl 1344.17003, MR 3381108, 10.15672/hjms.2015449110 |
| Reference:
|
[11] Cheng, Y., Su, Y.: (Co)homology and universal central extension of Hom-Leibniz algebras.Acta Math. Sin., Engl. Ser. 27 (2011), 813-830. Zbl 1250.17001, MR 2786445, 10.1007/s10114-011-9626-5 |
| Reference:
|
[12] Cheng, Y., Su, Y.: Quantum deformations of the Heisenberg-Virasoro algebra.Algebra Colloq. 20 (2013), 299-308. Zbl 1310.17007, MR 3043314, 10.1142/S1005386713000266 |
| Reference:
|
[13] Frégier, Y., Gohr, A., Silvestrov, S. D.: Unital algebras of Hom-associative type and surjective or injective twistings.J. Gen. Lie Theory Appl. 3 (2009), 285-295. Zbl 1237.17005, MR 2602991, 10.4303/jglta/S090402 |
| Reference:
|
[14] Gao, Y., Shang, S.: Universal coverings of Steinberg Lie algebras of small characteristic.J. Algebra. 311 (2007), 216-230. Zbl 1136.17016, MR 2309885, 10.1016/j.jalgebra.2006.10.044 |
| Reference:
|
[15] Hartwig, J. T., Larsson, D., Silvestrov, S. D.: Deformations of Lie algebras using $\sigma$-\hskip0ptderivations.J. Algebra 295 (2006), 314-361. Zbl 1138.17012, MR 2194957, 10.1016/j.jalgebra.2005.07.036 |
| Reference:
|
[16] Jin, Q., Li, X.: Hom-Lie algebra structures on semi-simple Lie algebras.J. Algebra 319 (2008), 1398-1408. Zbl 1144.17005, MR 2383052, 10.1016/j.jalgebra.2007.12.005 |
| Reference:
|
[17] Kassel, C., Loday, J.-L.: Central extensions of Lie algebras.Ann. Inst. Fourier 32 (1982), 119-142 French. Zbl 0485.17006, MR 0694130, 10.5802/aif.896 |
| Reference:
|
[18] Khalili, V.: On universal coverings of Lie tori.Bull. Korean Math. Soc. 49 (2012), 1199-1211. Zbl 1276.17013, MR 3002679, 10.4134/BKMS.2012.49.6.1199 |
| Reference:
|
[19] Li, X.: Representations of 3-dimensional simple multiplicative Hom-Lie algebras.Adv. Math. Phys. 2013 Article ID 938901, 7 pages. Zbl 1291.17010, MR 3132688, 10.1155/2013/938901 |
| Reference:
|
[20] Makhlouf, A., Silvestrov, S.: Hom-algebra structures.J. Gen. Lie Theory Appl. 2 (2008), 51-64. Zbl 1184.17002, MR 2399415, 10.4303/jglta/S070206 |
| Reference:
|
[21] Makhlouf, A., Silvestrov, S.: Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras.Forum Math. 22 (2010), 715-739. Zbl 1201.17012, MR 2661446, 10.1515/FORUM.2010.040 |
| Reference:
|
[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). Zbl 0874.17026, MR 1323858 |
| Reference:
|
[23] Neeb, K.-H.: Universal central extensions of Lie groups.Acts Appl. Math. 73 (2002), 175-219. Zbl 1019.22011, MR 1926500, 10.1023/A:1019743224737 |
| Reference:
|
[24] Neher, E.: An introduction to universal central extensions of Lie superalgebras.Groups, Rings, Lie and Hopf Algebras, 2001 Math. Appl. 555, Kluwer Academic Publishers, Dordrecht Y. Bahturin (2003), 141-166. Zbl 1077.17016, MR 1995057, 10.1007/978-1-4613-0235-3_10 |
| Reference:
|
[25] Neher, E., Sun, J.: Universal central extensions of direct limits of Lie superalgebras.J. Algebra 368 (2012), 169-181. Zbl 1301.17007, MR 2955226, 10.1016/j.jalgebra.2012.06.020 |
| Reference:
|
[26] Sheng, Y.: Representations of Hom-Lie algebras.Alger. Represent. Theory 15 (2012), 1081-1098. Zbl 1294.17001, MR 2994017, 10.1007/s10468-011-9280-8 |
| Reference:
|
[27] Kallen, W. L. J. van der: Infinitesimally Central Extensions of Chevalley Groups.Lecture Notes in Mathematics 356, Springer, Berlin (1973). Zbl 0275.17006, MR 0364484, 10.1007/BFb0060175 |
| Reference:
|
[28] Weibel, C. A.: An Introduction to Homological Algebra.Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge (1994). Zbl 0797.18001, MR 1269324, 10.1017/CBO9781139644136 |
| Reference:
|
[29] Yau, D.: Enveloping algebras of Hom-Lie algebras.J. Gen. Lie Theory Appl. 2 (2008), 95-108. Zbl 1214.17001, MR 2399418, 10.4303/jglta/S070209 |
| Reference:
|
[30] Yau, D.: Hom-algebras and homology.J. Lie Theory 19 (2009), 409-421. Zbl 1252.17002, MR 2572137 |
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