Low-dimensional cohomology of q-deformed Heisenberg-Virasoro algebra of Hom-type

Abstract

Hom-Lie algebras were introduced by J. Hartwig, D. Larsson, and S. Silvestrov as a generalized Lie algebra. When studying the homology and cohomology theory of Hom-Lie algebras, the authors find that the low-dimensional cohomology theory of Hom-Lie algebras is not well studied because of the Hom-Jacobi identity. In this paper, the authors compute the first and second cohomology groups of the q-deformed Heisenberg-Virasoro algebra of Hom-type, which will be useful to build the low-dimensional cohomology theory of Hom-Lie algebras.

Keywords

Hom-Lie algebra / q-deformed Heisenberg-Virasoro algebra of Hom-type / derivation / cohomology group

References

[1] Arbarello E., De Concini C., Kac V. G., Procesi C. Moduli spaces of curves and representation theory. Comm Math Phys, 1988, 117: 1-36.

[2] Farnsteiner R. Derivations and central extensions of finitely generated graded Lie algebra. J Algebra, 1988, 118: 33-45.

[3] Gao Y. The second Leibniz homology group for Kac-Moody Lie algebras. Bull of LMS, 2000, 32: 25-33.

[4] Hartwig J., Larsson D., Silvestrov S. Deformations of Lie algebras using σ-derivation. J Algebra, 2006, 295: 314-361.

[5] Jiang Q., Jiang C. Representations of the twisted Heisenberg-Virasoro algebra and the full Toroidal Lie Algebras. Algebra Colloq, 2007, 14(1): 117-134.

[6] Jin Q., Li X. Hom-Lie algebra structures on semisimple Lie algebras. J Algebra, 2008, 319: 1398-1408.

[7] Kac V., Cheung P. Quantum Calculus, 2002, Berlin: Springer-Verlag.

[8] Kassel C. Cyclic homology of differential operators, the Virasoro algebra and a q-analogue. Comm Math Phys, 1992, 146: 343-351.

[9] Larsson D., Silvestrov S. D. Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities. J Algebra, 2005, 288: 321-344.

[10] Leger G. F. Generalized derivations of Lie algebras. J Algebra, 2000, 228: 165-203.

[11] Li W. 2-Cocycles on the algebra of differential operators. J Algebra, 1989, 122: 64-80.

[12] Liu D., Zhu L. Generalized Heisenberg-Virasoro algebra. Front Math China, 2009, 4(2): 297-310.

[13] Loday J. -L., Pirashvili T. Universal enveloping algebras of Leibniz algebras and (co)homology. Math Ann, 1993, 296: 139-158.

[14] Makhlouf A, Silvestrov S D. Notes on formal deformations of Hom- associative and Hom-Lie algebras. arXiv:0712.3130v1 [math.RA]

[15] Scheunert M., Zhang R. B. Cohomology of Lie superalgebras and their generalizations. J Math Phys, 1998, 39: 5024-5061.

[16] Shen R., Jiang C. The derivation algebra and automorphism group of the twisted Heisenberg-Virasoro algebra. Comm Algebra, 2006, 34(7): 2547-2558.

[17] Su Y. 2-cocycles on the Lie algebras of generalized differential operators. Comm Algebra, 2002, 30: 763-782.

[18] Su Y. Low dimensional cohomology of general conformal algebras gc_N. J Math Phys, 2004, 45: 509-524.

[19] Su Y., Zhao K. Second cohomology group of generalized Witt type Lie algebras and certain representations. Comm Alegrba, 2002, 30: 3285-3309.

[20] Yau D. Hom-algebras as deformations and homology. J Lie Theory, 2009, 19: 409-421.

[21] Zhang Q., Zhang Y. Derivations and extensions of Lie color algebra. Acta Math Scientia, 2008, 28B(4): 933-948.