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Representation theory of the vertex algebra $W\sb{1+\infty}$. (English) Zbl 0862.17023
Representations of the (unique) central extension $\widehat {\cal D}$ 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 [{\it V. Kac} and {\it A. Radul}, Commun. Math. Phys. 157, 429-457 (1993; Zbl 0826.17027)] and [{\it E. Frenkel}, {\it V. Kac}, {\it A. Radul} and {\it 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 $c\in \bbfC$ there is an induced $\widehat {\cal D}$-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 $\widehat {\cal D}$ 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 $\widehat {\cal D}$ the representation $M_c$ is already irreducible for $c \notin \bbfZ$. The representations with $c= N \in \bbfN_0$ were studied in the above mentioned articles. The case $c = -N$, $N\in \bbfN$ 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.

17B69Vertex operators; vertex operator algebras and related structures
17B68Virasoro and related algebras
81T40Two-dimensional field theories, conformal field theories, etc.
17B10Representations of Lie algebras, algebraic theory
17B70Graded Lie (super)algebras
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