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Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle. (English) Zbl 0999.17032

From the introduction: The study of representation theory of the Lie algebra $\stackrel{^}{𝒟}$ (the universal central extension of the Lie algebra of differential operators on the circle, also denoted by ${W}_{1+\infty }$) was initiated in [V. Kac and 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 $\stackrel{^}{𝒟}$, 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 E. Frenkel, V. Kac, A. Radul, and W. Wang, Commun. Math. Phys. 170, 337-357 (1995; Zbl 0838.17028) and V. Kac and 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 $\stackrel{^}{{𝒟}^{M}}$ of the Lie algebra of $M×M$-matrix differential operators on the circle, its parabolic subalgebras, and the relation with $\stackrel{^}{gl}\left(\infty ,{R}_{m}\right)$. In Sections IV and V we classify and construct irreducible quasifinite highest weight modules over $\stackrel{^}{{𝒟}^{M}}$ and classify the unitary ones.

We consider the simple vertex algebra ${W}_{1+\infty ,c}^{M}$ constructed on the irreducible vacuum module of $\stackrel{^}{{𝒟}^{M}}$, and construct a large family of representations of this vertex algebra using twisted modules over $MN$ free charged fermions, proving thereby that all primitive $\stackrel{^}{{𝒟}^{M}}$-modules are vertex algebra modules for $c\in {ℤ}_{+}$. We conjecture that these are all irreducible modules over the vertex algebra ${W}_{1+\infty ,c}^{M}$ if $c$ is a positive integer.

##### MSC:
 17B66 Lie algebras of vector fields and related (super)algebras 17B10 Representations of Lie algebras, algebraic theory 17B69 Vertex operators; vertex operator algebras and related structures