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

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.


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
Full Text: DOI arXiv


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.