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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 𝒟 ^ (the universal central extension of the Lie algebra of differential operators on the circle, also denoted by W 1+ ) 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 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 𝒟 M ^ of the Lie algebra of M×M-matrix differential operators on the circle, its parabolic subalgebras, and the relation with gl ^(,R m ). In Sections IV and V we classify and construct irreducible quasifinite highest weight modules over 𝒟 M ^ and classify the unitary ones.

We consider the simple vertex algebra W 1+,c M constructed on the irreducible vacuum module of 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 𝒟 M ^-modules are vertex algebra modules for c + . We conjecture that these are all irreducible modules over the vertex algebra W 1+,c M if c is a positive integer.

17B66Lie algebras of vector fields and related (super)algebras
17B10Representations of Lie algebras, algebraic theory
17B69Vertex operators; vertex operator algebras and related structures