Second cohomology group of extended W-algebras

Abstract

Let \mathfrak{F} be a field of characteristic 0, and let G be an additive subgroup of \mathfrak{F}. We define a class of infinite-dimensional Lie algebras W with an \mathfrak{F}-basis \{L_{\mu },V_{\mu },W_{\mu }|\mu \in G\}, which are very closely related to W-algebras. In this paper, the second cohomology group of W is determined.

Keywords

W-algebra / second cohomology group / central extension

References

[1] Djokovic D, Zhao K. Derivations, isomorphisms and second cohomology of generalized Witt algebras. Tran Amer Math Soc, 1998, 350: 643-664

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

[3] Frenkel I, Lepowsky J, Meurman A. Vertex Operator Algebras and the Monster. New York: Academic Press, 1988

[4] Gao S, Jiang C, Pei Y. Structure of the extended Schrödinger-Virasoro Lie algebra sv ˜. Alg Colloq, 2009, 4: 549-566

[5] Kac V G. Infinite Dimensional Lie Algebras. Cambridge: Cambridge University Press, 1990

[6] Li J, Su Y, Zhu L. 2-Cocycles of original deformative Schrodinger-Virasoro algebras. Sci China, Ser A-Math, 2008, 38: 841-850

[7] Song G, Su Y. Derivations and 2-cocycles of contact Lie algebras related to locally-finite derivations. Comm Alg, 2004, 32: 4613-4631

[8] Song G, Su Y. 2-cocycles on the Lie superalgebras of Weyl type. Comm Alg, 2005, 33: 2991-3007

[9] Su Y. Harish-Chandra modules of the intermediate series over the high rank Virasoro algebras and high rank super-Virasoro algebras. J Math Phys, 1994, 35: 2013-2023

[10] Su Y. Simple modules over the high rank Virasoro algebras. Comm Alg, 2001, 29: 2067-2080

[11] Su Y. 2-Cocycles on the Lie algebras of generalized differential operators. Comm Alg, 2002, 30: 763-782

[12] Su Y. Structure of Lie superalgebras of block type related to locally finite derivations. Comm Alg, 2003, 31: 1725-1751

[13] Su Y, Zhao K. Second cohomology group of generalized Cartan type Witt Lie algebras and central extensions. Comm Alg, 2002, 30: 3285-3309

[14] Zamolodchikov A B. Infinite additional symmetries in two-dimensional conformal quantum field theory. Theor Math Phys, 1986, 65: 1205-1213

[15] Zhang W, Dong C. W-Algebra W(2, 2) and the Vertex operator algebra L(12,0)⊗L(12,0). Comm Math Phys, 2009, 258: 991-1004

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