W_{1+\infty} and W(gl_N) with central charge N

Abstract

We study representations of the central extension of the Lie algebra of differential operators on the circle, the W-infinity algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central charge N has a canonical structure of an irreducible representation of the W-algebra W(gl_N) with the same central charge and that all irreducible representations of W(gl_N) with central charge N arise in this way. We also establish a duality between "integral" modules of W(gl_N) and finite-dimensional irreducible modules of gl_N, and conjecture their fusion rules.

29 pages, Latex, uses file amssym.def (a few remarks added, typos corrected)

References in corpus (8)

• W-symmetry in Conformal Field Theory
• Quasifinite highest weight modules over the Lie algebra of differential operators on the circle
• Infinite Symmetry in the Quantum Hall Effect
• Fermions in the Lowest Landau Level: Bosonization, $W_{\infty }$ Algebra, Droplets, Chiral Boson
• Eigensystem and Full Character Formula of the W_{1+infinity} Algebra with c=1
• Determinant Formulae of Quasi-Finite Representation of W_{1+\infty} Algebra at Lower Levels
• Free Fields and Quasi-Finite Representation of $W_{1+\infty }$ Algebra
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Cited by in corpus (99)

• The $n$-component KP hierarchy and representation theory
• Thermal Transport in Chiral Conformal Theories and Hierarchical Quantum Hall States
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• Logarithmic intertwining operators and W(2,2p-1)-algebras
• Cosets of affine vertex algebras inside larger structures
• Chiral Deformations of Conformal Field Theories
• Simple current extensions beyond semi-simplicity
• Affine orbifolds and rational conformal field theory extensions of W_{1+infinity}
• Constructing quantum vertex algebras
• W{1+ \infty} algebra, W_3 algebra, and Friedan-Martinec-Shenker bosonization
• Twisted Supergravity and Koszul Duality: A case study in AdS${}_3$
• On the center of quiver-Hecke algebras
• W-Constraints for the Total Descendant Potential of a Simple Singularity
• Ghost Systems: A Vertex Algebra Point of View
• On quantum vertex algebras and their modules
• Logarithmic intertwining operators and vertex operators
• Invariant theory and the Heisenberg vertex algebra
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Crystal Melting and Black Holes
• W(1+infinity) algebra as a symmetry behind AGT relation
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• Hamiltonian Formulation of the W-Infinity Minimal Models
• Holomorphic field realization of SH${}^c$ and quantum geometry of quiver gauge theories
• Commutative subalgebras from Serre relations
• Virasoro constraint for Nekrasov instanton partition function
• The structure of the Kac-Wang-Yan algebra
• Fusion rules and complete reducibility of certain modules for affine Lie algebras
• Numerical Study of Hierarchical Hall Edge States on the Disk Geometry
• Chiral Partition Functions of Quantum Hall Droplets
• Invariant subalgebras of affine vertex algebras
• A string field theoretical description of (p,q) minimal superstrings
• A Hilbert theorem for vertex algebras
• Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions
• Notes on the algebraic curves in (p,q) minimal string theory
• W-infinity Symmetry in the Quantum Hall Effect Beyond the Edge
• Screening operators and Parabolic inductions for Affine W-algebras (with an appendix by Shigenori Nakatsuka)
• Generalized MacMahon G(q) as q-deformed CFT Correlation Function
• Algebraic bosonization: the study of the Heisenberg and Calogero-Sutherland models
• Introduction to vertex operator algebras I
• Mathieu Moonshine and the Geometry of K3 Surfaces
• Higher rank isomonodromic deformations and W-algebras
• 2D Superconductivity: Classification of Universality Classes by Infinite Symmetry
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• Tau-functions as highest weight vectors for W_{1+infty} algebra
• W-algebras from Heisenberg categories
• Snapshots of Conformal Field Theory
• A note on S-dual basis in free fermion system
• Twist-field representations of W-algebras, exact conformal blocks and character identities
• The generalized Heisenberg-Virasoro algebra
• Symmetry and Integrability of Non-Singlet Sectors in Matrix Quantum Mechanics
• Relations on ${\overline{\mathcal{M}}}_{g,n}$ and the negative $r$-spin Witten conjecture
• Jain states on a torus: an unifying description
• Genus Two Zhu Theory for Vertex Operator Algebras
• $W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities
• Supermatrix models and multi ZZ-brane partition functions in minimal superstring theories
• Algebro-Geometric Solutions of the Generalized Virasoro Constraints
• RCFT extensions of W_{1+infinity} in terms of bilocal fields
• The Higher Spin Rectangle
• Trace of the twisted Heisenberg Category
• The Hopf algebra $Rep{U}_q{\widehat{gl}}_{\infty }$
• Generalized $\tilde{W}$ algebras
• Representation theory of the vertex algebra $W_{1+\infty }$
• Tau functions of the charged free bosons
• The Bloch-Okounkov correlation functions of classical type
• Correlation functions of charged free boson and fermion systems
• The Character of the Infinite Wedge Representation
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• Infinite Dimensional Topological-Holomorphic Symmetry in Three-Dimensions
• Certain generating subspaces for vertex operator algebras
• W-hairs of black holes in three dimensional spacetime
• Vertex Algebras According to Isaac Newton
• Quasifinite Representations of Classical Lie subalgebras of $W_{\infty ,p}$
• Twisted Cherednik spectrum as a $q,t$-deformation
• Twisted vertex operators and Bernoulli polynomials
• Representations of the vertex algebra $W_{1+\infty }$ with a negative integer central charge
• CMM formula as superintegrability property of unitary model
• Generalized q-deformed Correlation Functions as Spectral Functions of Hyperbolic Geometry
• Associating quantum vertex algebras to Lie algebra $\gl_{\infty}$
• Conformal subalgebras of lattice vertex algebras
• Fusion rings for degenerate minimal models
• On a category of $g{l}_{\infty }$-modules
• Vertex algebras generated by Lie algebras
• Classification of Quasifinite Modules over the Lie Algebras of Weyl Type
• Classification of quasifinite $W_{\infty }$-modules
• Modular Virasoro Vertex Algebras and Affine Vertex Algebras
• The Brylinski filtration for affine Kac-Moody algebras and representations of $\mathcal{W}$-algebras
• Duality in infinite dimensional Fock representations
• Uniformization of the Moduli Space of Pairs Consisting of a Curve and a Vector Bundle
• The Bloch-Okounkov correlation functions at higher levels
• Representations of Centrally-Extended Lie Algebras over Differential Operators and Vertex Algebras
• Irregular vertex algebras
• Classification of Quasifinite Modules over Lie Algebras of Matrix Differential Operators on the Circle
• Explicit field realizations of W algebras
• Etingof Trace, Path Hypergeometric Functions and Integrable Systems
• Correlation functions of strict partitions and twisted Fock spaces
• A new class of Matrix Models arising from the W-infinity Algebra
• Quarter-indices for basic ortho-symplectic corners
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