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Hom-Lie superalgebras and Hom-Lie admissible superalgebras. (English) Zbl 1258.17008

J. T. Hartwig, D. Larsson and S. D. Silvestrov [“Deformation of Lie algebras using \(\sigma\)-derivations”, J. Algebra 295, No. 2, 314–361 (2006; Zbl 1138.17012)] introduced the notion of Hom-Lie algebra. It consists of a vector space \(V\) together with a skew-symmetric bilinear map \([-,-]\) and an endomorphism \(\alpha\) of \(V\) such that the Hom-Jacobi identity holds, namely \[ [\alpha(x),[y,z]]+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]=0. \] Hom-Lie algebras have been studied extensively by Silvestrov and his collaborators. In the present paper, the authors introduce the super-version of Hom-Lie algebras, and study some of their properties.

MSC:

17A70 Superalgebras
16W55 “Super” (or “skew”) structure
17B68 Virasoro and related algebras

Citations:

Zbl 1138.17012
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References:

[1] Aizawa, N.; Sato, H., \(q\)-Deformation of the Virasoro algebra with central extension, Phys. Lett. B, 256, 1, 185-190 (1991) · Zbl 1332.17011
[2] Ataguema, H.; Makhlouf, A.; Silvestrov, S., Generalization of \(n\)-ary Nambu algebras and beyond, J. Math. Phys., 50, 1 (2009) · Zbl 1328.17004
[3] Caenepeel, S.; Goyvaerts, I., Monoidal Hom-Hopf algebras (2009) · Zbl 1255.16032
[4] Chaichian, M.; Isaev, A. P.; Lukierski, J.; Popowicz, Z.; Prešnajder, P., \(q\)-Deformations of Virasoro algebra and conformal dimensions, Phys. Lett. B, 262, 1, 32-38 (1991)
[5] Chaichian, M.; Kulish, P.; Lukierski, J., \(q\)-Deformed Jacobi identity, \(q\)-oscillators and \(q\)-deformed infinite-dimensional algebras, Phys. Lett. B, 237, 3-4, 401-406 (1990)
[6] Chaichian, M.; Popowicz, Z.; Prešnajder, P., \(q\)-Virasoro algebra and its relation to the \(q\)-deformed KdV system, Phys. Lett. B, 249, 1, 63-65 (1990)
[7] Frégier, Y.; Gohr, A., On Hom type algebras (2009)
[8] Goze, M.; Remm, E., Lie-admissible algebras and operads, J. Algebra, 273, 129-152 (2004) · Zbl 1045.17007
[9] Hartwig, J. T.; Larsson, D.; Silvestrov, S. D., Deformations of Lie algebras using \(σ\)-derivations, J. Algebra, 295, 314-361 (2006) · Zbl 1138.17012
[10] Hu, N., \(q\)-Witt algebras, \(q\)-Lie algebras, \(q\)-holomorph structure and representations, Algebra Colloq., 6, 1, 51-70 (1999) · Zbl 0943.17007
[11] Larsson, D.; Silvestrov, S. D., Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra, 288, 321-344 (2005) · Zbl 1099.17015
[12] Larsson, D.; Silvestrov, S. D., Quasi-Lie algebras, (Noncommutative Geometry and Representation Theory in Mathematical Physics. Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemp. Math., vol. 391 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 241-248 · Zbl 1105.17005
[13] Larsson, D.; Silvestrov, S. D., Quasi-deformations of \(sl_2(F)\) using twisted derivations, Comm. Algebra, 35, 12, 4303-4318 (2007) · Zbl 1131.17010
[14] Larsson, D.; Silvestrov, S. D., Graded quasi-Lie agebras, Czechoslovak J. Phys., 55, 1473-1478 (2005)
[15] Liu, K., Characterizations of quantum Witt algebra, Lett. Math. Phys., 24, 257-265 (1992) · Zbl 0759.17007
[16] Makhlouf, A.; Silvestrov, S. D., Hom-algebra structures, J. Gen. Lie Theory Appl., 2, 2, 51-64 (2008) · Zbl 1184.17002
[17] Makhlouf, A.; Silvestrov, S. D., Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, (Silvestrov, S.; etal., Generalized Lie Theory in Mathematics, Physics and Beyond (2008), Springer-Verlag: Springer-Verlag Berlin), 189-206, (Chapter 17) · Zbl 1173.16019
[18] Makhlouf, A.; Silvestrov, S. D., Notes on formal deformations of Hom-associative and Hom-Lie algebras (2007), Forum Math., in press, Preprints in Math. Sci., Lund Univ., Center for Math. Sci., 2007
[19] Makhlouf, A.; Silvestrov, S. D., Hom-algebras and Hom-coalgebras (2008), J. Algebra Appl., in press, Preprints in Math. Sci., Lund Univ., Center for Math. Sci., 2008 · Zbl 1259.16041
[20] Vinberg, E. B., Convex homogeneous cones, Transl. Moscow Math. Soc., 12, 340-403 (1963) · Zbl 0138.43301
[21] Yau, D., Enveloping algebra of Hom-Lie algebras, J. Gen. Lie Theory Appl., 2, 2, 95-108 (2008) · Zbl 1214.17001
[22] Yau, D., Hom-algebras and homology, J. Lie Theory, 19, 409-421 (2009) · Zbl 1252.17002
[23] Yau, D., Hom-bialgebras and comodule algebras, Int. Electron. J. Algebra (2008), in press · Zbl 1253.16032
[24] Yau, D., Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras, J. Phys. A, 42, 165-202 (2009) · Zbl 1179.17001
[25] Yau, D., The Hom-Yang-Baxter equation and Hom-Lie algebras (2009) · Zbl 1179.17001
[26] Yau, D., The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras (2009) · Zbl 1179.17001
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