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Representation theory of the vertex algebra W 1+ . (English) Zbl 0862.17023

Representations of the (unique) central extension 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 gl ^() of the Lie algebra of infinite matrices with only finitely many diagonals and with the help of vertex algebras. For every central charge c there is an induced 𝒟 ^-module M c . This module admits a canonical vertex algebra structure. The unique irreducible quotient is denoted by W 1+ . The highest weight representations of the vertex algebra M c are in canonical 1-1 correspondence to the highest weight representations of 𝒟 ^ with central charge c. The goal is to describe the representations of the simple vertex algebra W 1+ . By suitable normalisation of the defining cocycle for 𝒟 ^ the representation M c is already irreducible for c. The representations with c=N 0 were studied in the above mentioned articles. The case c=-N, 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 gl ^(). 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:
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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