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Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities. (English) Zbl 1099.17015
The authors first introduce the notion of quasi-hom-Lie algebras, a natural generalization of hom-Lie algebras introduced by J. T. Hartwig and the authors [J. Algebra 295, No. 2, 314–361 (2006; Zbl 1138.17012)]. Quasi-hom-Lie algebras include color Lie algebras and superalgebras, and can be seen as deformations of these by maps, twisting the Jacobi identity and skew-symmetry. The goal for introducing the quasi-hom-Lie algebras is to generalize or deform the Witt algebra of derivations on the Laurent polynomials \({\mathbb C}[t,t^{-1}]\). A theory of central extensions for quasi-hom-Lie algebras is developed. The main result of the paper is the description of central extensions of quasi-hom-Lie algebras in terms of equivalence classes of 2-cocycle-like maps.
Reviewer: Yucai Su (Hefei)

MSC:
17B61 Hom-Lie and related algebras
17B56 Cohomology of Lie (super)algebras
17B68 Virasoro and related algebras
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References:
[1] Aizawa, N.; Sato, H.-T., q-deformation of the Virasoro algebra with central extension, Phys. lett. B, 256, 2, 185-190, (1999) · Zbl 1332.17011
[2] Bloch, S., Zeta values and differential operators on the circle, J. algebra, 182, 476-500, (1996) · Zbl 0868.17017
[3] 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)
[4] 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)
[5] Chaichian, M.; Prešnajder, P., q-Virasoro algebra, q-conformal dimensions and free q-superstring, Nuclear phys. B, 482, 1-2, 466-478, (1996) · Zbl 0974.81507
[6] Chung, W.-S., Two parameter deformation of Virasoro algebra, J. math. phys., 35, 5, 2490-2496, (1994) · Zbl 0822.17015
[7] Curtright, T.L.; Zachos, C.K., Deforming maps for quantum algebras, Phys. lett. B, 243, 3, 237-244, (1990)
[8] ()
[9] Di Francesco, P.; Mathieu, P.; Sénéchal, D., Conformal field theory, ISBN: 0-387-94785-X, (1997), Springer, 890 pp · Zbl 0869.53052
[10] Frenkel, I.; Lepowsky, J.; Meurman, A., Vertex operator algebras and the monster, ISBN: 0-12-267065-5, (1988), Academic Press, 508 pp · Zbl 0674.17001
[11] Fuchs, J., Affine Lie algebras and quantum groups, (1992), Cambridge University Press, 433 pp
[12] Fuchs, J., Lectures on conformal field theory and kac – moody algebras, Springer lecture notes in phys., vol. 498, (1997), pp. 1-54 · Zbl 0920.17012
[13] Fuks, D.B., Cohomology of infinite-dimensional Lie algebras, ISBN: 0-306-10990-5, (1986), Plenum · Zbl 0667.17005
[14] Hartwig, J.T.; Larsson, D.; Silvestrov, S.D., Deformations of Lie algebras using σ-derivations, Preprints in Mathematical Sciences 2003:32, LUTFMA-5036-2003, Centre for Mathematical Sciences, Department of Mathematics, Lund Institute of Technology, Lund University, 2003, preprint
[15] Hellström, L.; Silvestrov, S.D., Commuting elements in q-deformed Heisenberg algebras, ISBN: 981-02-4403-7, (2000), World Scientific, 256 pp · Zbl 0956.17006
[16] H.P. Jakobsen, Matrix chain models and their q-deformations, Preprint Mittag-Leffler Institute, Report Nr. 23, 2003/2004, ISSN 1103-467X, ISRN IML-R-23-03/04-SE
[17] Jakobsen, H.P.; Lee, H.C.-W., Matrix chain models and kac – moody algebras, (), 147-165 · Zbl 1142.17307
[18] Kac, V.G., Simple irreducible graded Lie algebras of finite growth, Math. USSR izv., 2, 1271-1311, (1968) · Zbl 0222.17007
[19] Kac, V.G.; Raina, A.K., Highest weight representations of infinite-dimensional Lie algebras, ISBN: 9971-50-395-6, (1987), World Scientific, 145 pp · Zbl 0668.17012
[20] Kassel, C., Cyclic homology of differential operators, the Virasoro algebra and a q-analogue, Comm. math. phys., 146, 343-351, (1992) · Zbl 0761.17020
[21] Khesin, B.; Lyubashenko, V.; Roger, C., Extensions and contractions of Lie algebra of q-pseudodifferential symbols on the circle, J. func. anal., 143, 55-97, (1997) · Zbl 0872.35138
[22] Larsson, D.; Silvestrov, S.D., Quasi-Hom-Lie algebras, central extensions and 2-cocycle-like identities, Preprints in Mathematical Sciences 2004:3, ISSN 1403-9338, LUTFMA-5038-2004
[23] Li, W.-L., 2-cocycles on the algebra of differential operators, J. algebra, 122, 64-80, (1989) · Zbl 0671.17010
[24] Moody, R.V., A new class of Lie algebras, J. algebra, 10, 211-230, (1968) · Zbl 0191.03005
[25] Passman, D.S., Simple Lie color algebras of Witt type, J. algebra, 208, 698-721, (1998) · Zbl 0923.17013
[26] Polychronakos, A.P., Consistency conditions and representations of a q-deformed Virasoro algebra, Phys. lett. B, 256, 1, 35-40, (1991) · Zbl 1332.81097
[27] Sato, H.-T., Realizations of q-deformed Virasoro algebra, Progr. theoret. phys., 89, 2, 531-544, (1993)
[28] Sato, H.-T., q-Virasoro operators from an analogue of the Noether currents, Z. phys. C, 70, 2, 349-355, (1996)
[29] Scheunert, M., Introduction to the cohomology of Lie superalgebras and some applications, Res. exp. math., 25, 77-107, (2002) · Zbl 1022.17013
[30] Scheunert, M.; Zhang, R.B., Cohomology of Lie superalgebras and their generalizations, J. math. phys., 39, 5024-5061, (1998) · Zbl 0928.17023
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