*(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, algebraic theory |

17B69 | Vertex operators; vertex operator algebras and related structures |