Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle.

*(English)*Zbl 0999.17032From 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.

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 |

##### Keywords:

universal central extension; Lie algebra of differential operators on the circle; \(W_{1+\infty}\); simple vertex algebra
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\textit{C. Boyallian} et al., J. Math. Phys. 39, No. 5, 2910--2928 (1998; Zbl 0999.17032)

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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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