Representation theory of the vertex algebra \(W_{1+\infty}\). (English) Zbl 0862.17023

Kac, Victor; Radul, Andrey

Transform. Groups 1, No. 1-2, 41-70 (1996).

Representations of the (unique) central extension \(\widehat {\mathcal D}\) of the Lie algebra of differential operators (with a finite number of Fourier modes) on the circle are studied. The authors together with other collaborators began these studies in the articles [V. Kac and A. Radul, Commun. Math. Phys. 157, 429-457 (1993; Zbl 0826.17027)] and [E. Frenkel, V. Kac, A. Radul and W. Wang, Commun. Math. Phys. 170, 337-357 (1995; Zbl 0838.17028)]. The representations are studied with the help of irreducible highest weight representations of the central extension \(\widehat {gl} (\infty)\) of the Lie algebra of infinite matrices with only finitely many diagonals and with the help of vertex algebras. For every central charge \(c\in \mathbb{C}\) there is an induced \(\widehat {\mathcal D}\)-module \(M_c\). This module admits a canonical vertex algebra structure. The unique irreducible quotient is denoted by \(W_{1+ \infty}\). The highest weight representations of the vertex algebra \(M_c\) are in canonical \(1-1\) correspondence to the highest weight representations of \(\widehat {\mathcal D}\) with central charge \(c\). The goal is to describe the representations of the simple vertex algebra \(W_{1+ \infty}\). By suitable normalisation of the defining cocycle for \(\widehat {\mathcal D}\) the representation \(M_c\) is already irreducible for \(c \notin \mathbb{Z}\). The representations with \(c= N \in \mathbb{N}_0\) were studied in the above mentioned articles. The case \(c = -N\), \(N\in \mathbb{N}\) is considered in the article under review.
The main result is a decomposition of the vertex algebra of \(N\) charged free bosons with respect to the commuting pair of Lie algebras \(gl(N)\) and \(\widehat {gl} (\infty)\). In this way a large class of irreducible modules are produced. The authors conjecture that all irreducible modules can be obtained by applying certain constructions to them. Explicit character formulas are given. A basic tool is a modified theory of dual Howe pairs.

Reviewer: M.Schlichenmaier (Mannheim)

MSC:

17B69	Vertex operators; vertex operator algebras and related structures
17B68	Virasoro and related algebras
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B70	Graded Lie (super)algebras

Keywords:

Verma modules; central extension; Lie algebra of differential operators; highest weight representations; vertex algebras

Citations:

Zbl 0826.17027; Zbl 0838.17028

References:

[1]	[AFMO] H. Awata, M. Fukuma, Y. Matsuo, and S. Odake,Character and determinant formulae of quasifinite representations of the W 1+algebra, preprint, RIMS-982 (1994). · Zbl 0853.17023
[2]	[ABMNV] L. Alvarez-Gaumé, J.B. Bost, G. Moore, P. Nelson, and C. Vafa,Bosonization in arbitrary genus, Phys. Lett. B178 (1986), 41–45.
[3]	[ASM] M. Adler, T. Shiota, and P. van Moerbeke,From the w algebra to its central extension: A {\(\tau\)}-function approach, preprint. · Zbl 0925.58031
[4]	[BMP] P. Bouwknegt, J. McCarthy, and K. Pilch,Semi-infinite cohomology and w-gravity, hep-th 9302086. · Zbl 0804.53100
[5]	[B1] R. Borcherds,Vertex algebras, Kač-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA83 (1986) 3068–3071. · Zbl 0613.17012
[6]	R. Borcherds,Monstrous moonshine and monstrous Lie superalgebras, Invent. Math.109 (1992), 405–444. · Zbl 0799.17014
[7]	[CTZ1] A. Capelli, C.A. Trugenberger, and G.R. Zemba,Classifications of quantum Hall universality classes by W 1+symmetry, Phys. Rev. Let.72 (1994), 1902–1905.
[8]	[CTZ2] A. Capelli, C.A. Trugenberger, and G.R. Zemba,Stable hierarchical quantum Hall fluids as W 1+minimal models, preprint, UGVA-DPT 1995/01-879.
[9]	[D] J. Dixmier,Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. · Zbl 0308.17007
[10]	[F] B.L. Feigin,The Lie algebra gl() and the cohomology of the Lie algebra of differential operators, Uspechi Math. Nauk35, no. 2 (1988), 157–158. · Zbl 0653.17009
[11]	[FKRW] E. Frenkel, V. Kac, A. Radul, and W. Wang,W 1+and W(glN) with central charge N, Comm. Math. Phys.170 (1995), 337–357. · Zbl 0838.17028
[12]	[FLM] I.B. Frenkel, J. Lepowsky, and A. Meurman,Vertex Operator Algebras and the Monster, Academic Press, New York, 1988. · Zbl 0674.17001
[13]	[G] P. Goddard,Meromorphic conformal field theory, in Infinite-dimensional Lie algebras and groups (V. Kac, ed.) Adv. Ser. in Math. Phys7 (1989), World Scientific, 556–587. · Zbl 0742.17027
[14]	R. Howe,Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons, Lectures in Appl. Math., vol. 21, 1985, pp. 179–206. · Zbl 0558.22018
[15]	[H2] R. Howe,Remarks on classical invariant theory, Trans. AMS313 (1989), 539–570. · Zbl 0674.15021
[16]	[K1] V.G. Kac,Infinite-Dimensional Lie Algebras, third edition, Cambridge University Press, 1990. · Zbl 0716.17022
[17]	[K2] V.G. Kac,Vertex algebras preprint.
[18]	[KP] V.G. Kac and D.H. Peterson,Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA78 (1981), 3308–3312. · Zbl 0469.22016
[19]	[KR] V.G. Kac and A. Radul,Quasi-finite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys.157 (1993), 429–457. · Zbl 0826.17027
[20]	[KV] M. Kashiwara and M. Vergne,On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math.44 (1978), 1–47. · Zbl 0375.22009
[21]	[L] S. Lang,Algebra, Addison-Wesley Publ. Co., Reading, MA, 1965.
[22]	[Li] H. Li,2-cocycles on the algebra of differential operators, J. Algebra122 (1989), 64–80. · Zbl 0671.17010
[23]	[M] I. Macdonald,Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. · Zbl 0487.20007
[24]	[Mat] Y. Matsuo,Free fields and quasi-finite representations of W 1 Phys. Lett. B.326 (1994), 95–100. · Zbl 0894.17019
[25]	[O] G.I. Olshanskii,Description of the representations of U(p,q) with highest weight, Funct. Anal. Appl14 (1980), 32–44.
[26]	[PV] V.L. Popov and E.B. Vinberg,Invariant Theory, Encyclopaedia of Math. Sci.: Algebraic Geometry IV, vol. 55, Springer-Verlag, Berlin, Heidelberg, and New-York, 1994, pp. 123–284.
[27]	[Z] D.P. Zhelobenko,Compact Lie groups and their representations, Nauka, Moscow, 1970. · Zbl 0228.22013