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Zbl 0999.17032
Boyallian, Carina; Kac, Victor G.; Liberati, José I.; Yan, Catherine H.
Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle. (English)
[J] J. Math. Phys. 39, No.5, 2910-2928 (1998). ISSN 0022-2488; ISSN 1089-7658/e

From the introduction: The study of representation theory of the Lie algebra $\widehat{\cal D}$ (the universal central extension of the Lie algebra of differential operators on the circle, also denoted by $W_{1+\infty}$) was initiated in [{\it V. Kac} and {\it A. Radul}, Commun. Math. Phys. 157, 429-457 (1993; Zbl 0826.17027)]. In that paper, Kac and Radul classified the irreducible quasifinite highest weight representations of $\widehat{\cal D}$, realized them in terms of irreducible highest weight representations of the Lie algebra of infinite matrices, and described the unitary ones. This study was continued [in {\it E. Frenkel, V. Kac, A. Radul}, and {\it W. Wang}, Commun. Math. Phys. 170, 337-357 (1995; Zbl 0838.17028) and {\it V. Kac} and {\it A. Radul}, Transform. Groups 1, 41-70 (1996; Zbl 0862.17023)] in the framework of vertex algebra theory. It was mentioned at the end of the first Kac-Radul paper that similar results can be obtained in the matrix case, and this is the main goal of the present paper. We study the structure of the central extension $\widehat{{\cal D}^M}$ of the Lie algebra of $M\times M$-matrix differential operators on the circle, its parabolic subalgebras, and the relation with $\widehat{gl} (\infty,R_m)$. In Sections IV and V we classify and construct irreducible quasifinite highest weight modules over $\widehat{{\cal D}^M}$ and classify the unitary ones. \par We consider the simple vertex algebra $W_{1+\infty,c}^M$ constructed on the irreducible vacuum module of $\widehat{{\cal D}^M}$, and construct a large family of representations of this vertex algebra using twisted modules over $MN$ free charged fermions, proving thereby that all primitive $\widehat{{\cal D}^M}$-modules are vertex algebra modules for $c\in \bbfZ_+$. We conjecture that these are all irreducible modules over the vertex algebra $W_{1+\infty,c}^M$ if $c$ is a positive integer.

MSC 2000:: *17B66 Lie algebras of vector fields and related algebras
17B10 Representations of Lie algebras, algebraic theory
17B69 Vertex operators

Keywords: universal central extension; Lie algebra of differential operators on the circle; $W_{1+\infty}$; simple vertex algebra

Citations: Zbl 0838.17028; Zbl 0826.17027; Zbl 0862.17023

Cited in: Zbl 1009.17017

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