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

Representations of the (unique) central extension $\stackrel{^}{𝒟}$ 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 $\stackrel{^}{gl}\left(\infty \right)$ of the Lie algebra of infinite matrices with only finitely many diagonals and with the help of vertex algebras. For every central charge $c\in ℂ$ there is an induced $\stackrel{^}{𝒟}$-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 $\stackrel{^}{𝒟}$ 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 $\stackrel{^}{𝒟}$ the representation ${M}_{c}$ is already irreducible for $c\notin ℤ$. The representations with $c=N\in {ℕ}_{0}$ were studied in the above mentioned articles. The case $c=-N$, $N\in ℕ$ 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 $gl\left(N\right)$ and $\stackrel{^}{gl}\left(\infty \right)$. 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.

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