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Representations of hom-Lie algebras. (English) Zbl 1294.17001
This paper provides a framework for the representation theory of hom-Lie algebras: the author defines the \(\alpha^k\)-derivation of a multiplicative hom-Lie algebra and considers the corresponding derivation extension; he defines the representation of a multiplicative hom-Lie algebra and the corresponding hom-cochain complexes and coboundary operators explicitly; he shows that central extensions of a multiplicative hom-Lie algebra are controlled by the second cohomology with coefficients in the trivial representation; he also studies the adjoint representations of a regular hom-Lie algebra; finally, he defines the hom-Nijenhuis operator of a regular hom-Lie algebra, which could give a trivial deformation.

MSC:
17A30 Nonassociative algebras satisfying other identities
17B99 Lie algebras and Lie superalgebras
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[1] Ammar, F., Ejbehi, Z., Makhlouf, A.: Cohomology and Deformations of Hom-algebras. arXiv:1005.0456 (2010) · Zbl 1237.17003
[2] Benayadi, S., Makhlouf, A.: Hom-Lie Algebras with Symmetric Invariant NonDegenerate Bilinear Forms. arXiv:1009.4226 (2010) · Zbl 1331.17028
[3] Chevalley, C., Eilenberg, S.: Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63, 85–124 (1948) · Zbl 0031.24803
[4] Dorfman, I.: Dirac Structures and Integrability of Nonlinear Evolution Equation. Wiley, New York (1993)
[5] Hartwig, J., Larsson, D., Silvestrov, S.: Deformations of Lie algebras using {\(\sigma\)}-derivations. J. Algebra 295, 314–361 (2006) · Zbl 1138.17012
[6] Jacobson, N.: Lie Algebras. Dover Publications, Inc. New York (1962) · Zbl 0121.27504
[7] Larsson, D., Silvestrov, S.: Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities. J. Algebra 288, 321–344 (2005) · Zbl 1099.17015
[8] Larsson, D., Silvestrov, S.: Quasi-Lie algebras. Contemp. Math. 391, 241–248 (2005) · Zbl 1105.17005
[9] Makhlouf, A., Silvestrov, S.: Notes on formal deformations of hom-associative and hom-Lie algebras. Forum Math. 22(4), 715–739 (2010) · Zbl 1201.17012
[10] Makhlouf, A., Silvestrov, S.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008) · Zbl 1184.17002
[11] Sheng, Y.: Linear Poisson structures on $\(\backslash\)mathbb R\^4$ . J. Geom. Phys. 57, 2398–2410 (2007) · Zbl 1131.53045
[12] Yau, D.: Hom–Yang–Baxter equation, hom-Lie algebras, and quasi-triangular bialgebras. J. Phys. A: Math. Theory 42, 165202 (2009) · Zbl 1179.17001
[13] Yau, D.: Hom-algebras and homology. J. Lie Theory 19, 409–421 (2009) · Zbl 1252.17002
[14] Yau, D.: Enveloping algebras of hom-Lie algebras. J. Gen. Lie Theory Appl. 2, 95–108 (2008) · Zbl 1214.17001
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