Second cohomology group of extended W-algebras

Abstract

Let $\mathbb{F}$ be a field of characteristic 0, and let G be an additive subgroup of $\mathbb{F}$. We define a class of infinite-dimensional Lie algebras [inline-graphic not available: see fulltext] with an $\mathbb{F}$-basis {L _µ, V _µ, W _µ | µ ∈ G}, which are very closely related to W-algebras. In this paper, the second cohomology group of [inline-graphic not available: see fulltext] 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, 1988, New York: Academic Press.

[4.] Gao S., Jiang C., Pei Y. Structure of the extended Schrödinger-Virasoro Lie algebra $\widetilde{\mathfrak{s}\mathfrak{v}}$. Alg Colloq, 2009, 4, 549-566

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

[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 $L\left(\frac{1}{2},0\right)\otimes L\left(\frac{1}{2},0\right)$. Comm Math Phys, 2009, 258, 991-1004