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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 $$\widehat{\mathcal D}$$ (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 $$\widehat{\mathcal D}$$, 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 $$\widehat{{\mathcal D}^M}$$ 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 $$\widehat{{\mathcal D}^M}$$ and classify the unitary ones.
We consider the simple vertex algebra $$W_{1+\infty,c}^M$$ constructed on the irreducible vacuum module of $$\widehat{{\mathcal D}^M}$$, and construct a large family of representations of this vertex algebra using twisted modules over $$MN$$ free charged fermions, proving thereby that all primitive $$\widehat{{\mathcal D}^M}$$-modules are vertex algebra modules for $$c\in \mathbb{Z}_+$$. 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 and Lie superalgebras, algebraic theory (weights) 17B69 Vertex operators; vertex operator algebras and related structures
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References:
 [1] DOI: 10.1007/BF02096878 · Zbl 0826.17027 · doi:10.1007/BF02096878 [2] DOI: 10.1007/BF02108332 · Zbl 0838.17028 · doi:10.1007/BF02108332 [3] DOI: 10.1073/pnas.78.6.3308 · Zbl 0469.22016 · doi:10.1073/pnas.78.6.3308 [4] DOI: 10.1073/pnas.83.10.3068 · Zbl 0613.17012 · doi:10.1073/pnas.83.10.3068
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