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Keywords:
Hom-Lie superalgebra; derivation; cohomology; $q$-deformed superalgebra
Hom-Lie superalgebra; derivation; cohomology; $q$-deformed superalgebra
Summary:
Hom-Lie algebra (superalgebra) structure appeared naturally in $q$-deformations, based on $\sigma $-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of $\alpha ^k$-derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted ${\rm osp}(1,2)$ superalgebra and $q$-deformed Witt superalgebra.
References:
[1] Aizawa, N., Sato, H.: $q$-deformation of the Virasoro algebra with central extension. Phys. Lett. B 256 (1991), 185-190. DOI 10.1016/0370-2693(91)90671-C | MR 1099971
[2] Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and deformations of Hom-algebras. J. Lie Theory 21 (2011), 813-836. MR 2917693 | Zbl 1237.17003
[3] Ammar, F., Makhlouf, A.: Hom-Lie superalgebras and Hom-Lie admissible superalgebras. J. Algebra 324 (2010), 1513-1528. DOI 10.1016/j.jalgebra.2010.06.014 | MR 2673748 | Zbl 1258.17008
[4] Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. Submitted to J. Geom. Phys.
[5] Hartwig, J. T., Larsson, D., Silvestrov, S. D.: Deformations of Lie algebras using $\sigma$-derivations. J. Algebra 295 (2006), 314-361. DOI 10.1016/j.jalgebra.2005.07.036 | MR 2194957 | Zbl 1138.17012
[6] Hu, N.: $q$-Witt algebras, $q$-Lie algebras, $q$-holomorph structure and representations. Algebra Colloq. 6 (1999), 51-70. MR 1680657 | Zbl 0943.17007
[7] Larsson, D., Silvestrov, S. D.: Quasi-hom-Lie algebras, central extensions and $2$-cocyclelike identities. J. Algebra 288 (2005), 321-344. DOI 10.1016/j.jalgebra.2005.02.032 | MR 2146132
[8] Larsson, D., Silvestrov, S. D.: Quasi-Lie algebras. Noncommutative Geometry and Representation Theory in Mathematical Physics. Satellite conference to the fourth European congress of mathematics, July 5-10, 2004, Karlstad, Sweden. Contemporary Mathematics 391 J. Fuchs American Mathematical Society Providence, RI (2005), 241-248. MR 2184027 | Zbl 1105.17005
[9] Larsson, D., Silvestrov, S. D.: Quasi-deformations of $sl_2(\mathbb{F})$ using twisted derivations. Commun. Algebra 35 (2007), 4303-4318. DOI 10.1080/00927870701545127 | MR 2372334
[10] Larsson, D., Silvestrov, S. D.: Graded quasi-Lie algebras. Czechoslovak J. Phys. 55 (2005), 1473-1478. DOI 10.1007/s10582-006-0028-3 | MR 2223838
[11] Liu, K.: Characterizations of quantum Witt algebra. Lett. Math. Phys. 24 (1992), 257-265. DOI 10.1007/BF00420485 | MR 1172453
[12] Makhlouf, A.: Paradigm of nonassociative Hom-algebras and Hom-superalgebras. Proceedings of Jordan Structures in Algebra and Analysis Meeting. Tribute to El Amin Kaidi for his 60th birthday, Almería, Spain, May 20-22, 2009 J. Carmona Tapia et al. Univ. de Almería, Departamento de Álgebra y Análisis Matemático Almería (2010), 143-177. MR 2648355 | Zbl 1252.17001
[13] Makhlouf, A., Silvestrov, S. D.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2 (2008), 51-64. DOI 10.4303/jglta/S070206 | MR 2399415 | Zbl 1184.17002
[14] Makhlouf, A., Silvestrov, S.: Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras. Generalized Lie Theory in Mathematics, Physics and Beyond S. Silvestrov et al. Springer Berlin (2009), 189-206. MR 2509148 | Zbl 1173.16019
[15] Makhlouf, A., Silvestrov, S. D.: Notes on $1$-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22 (2010), 715-759. DOI 10.1515/forum.2010.040 | MR 2661446 | Zbl 1201.17012
[16] Scheunert, M.: Generalized Lie algebras. J. Math. Phys. 20 (1979), 712-720. DOI 10.1063/1.524113 | MR 0529734 | Zbl 0423.17003
[17] Scheunert, M.: The Theory of Lie Superalgebras. An Introduction. Lecture Notes in Mathematics 716. Springer Berlin (1979). DOI 10.1007/BFb0070929 | MR 0537441
[18] Scheunert, M., Zhang, R. B.: Cohomology of Lie superalgebras and their generalizations. J. Math. Phys. 39 (1998), 5024-5061. DOI 10.1063/1.532508 | MR 1643330 | Zbl 0928.17023
[19] Sheng, Y.: Representations of Hom-Lie algebras. Algebr. Represent. Theory 15 (2012), 1081-1098. DOI 10.1007/s10468-011-9280-8 | MR 2994017
[20] Yau, D.: Enveloping algebras of Hom-Lie algebras. J. Gen. Lie Theory Appl. 2 (2008), 95-108. DOI 10.4303/jglta/S070209 | MR 2399418 | Zbl 1214.17001
[21] Yau, D.: Hom-algebras and homology. J. Lie Theory 19 (2009), 409-421. MR 2572137 | Zbl 1252.17002
[22] Yau, D.: Hom-bialgebras and comodule Hom-algebras. Int. Electron. J. Algebra 8 (2010), 45-64. MR 2660540 | Zbl 1253.16032
[23] Yau, D.: The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras. J. Phys. A, Math. Theor. 42 (2009), Article ID 165202, 12 pages. MR 2539278 | Zbl 1179.17001
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