*(English)*Zbl 0826.17027

In the article it is shown how the algebra of differential operators $\mathcal{D}$ on the circle, respectively its universal central extension $\widehat{\mathcal{D}}$, can be obtained via a twisted Laurent polynomial algebra over a polynomial algebra. In a similar way the algebras ${\mathcal{D}}_{q}$ of difference operators on the circle and their central extensions are obtained. They are related to the trigonometric Sin- algebras. The introduced algebras are graded algebras with the grading induced by Laurent polynomial part. In particular, the homogeneous parts are infinite-dimensional. Quasifinite highest weight modules of these algebras (considered as Lie algebras) are highest weight modules over them (they are graded) for which the homogeneous parts are finite- dimensional. Starting from a Verma module $M\left(\lambda \right)$ associated to the weight $\lambda $ one obtains the unique irreducible highest weight module $L\left(\lambda \right)$ as a quotient.

Now the authors introduce with respect to parabolic subalgebras generalized Verma modules. To such a parabolic subalgebra a characteristic polynomial is associated. In the differential operator algebra case they show the essential fact that $L\left(\lambda \right)$ is quasifinite if and only if $L\left(\lambda \right)$ is a quotient of a generalized Verma module $M(\lambda ,\U0001d52d)$ with respect to a parabolic subalgebra $\U0001d52d$ (or equivalently to a characteristic polynomial). The authors give necessary and sufficient conditions for $L\left(\lambda \right)$ to be a quasifinite irreducible highest weight module in terms of “labels” of the weight. For explicit constructions the homomorphism from these algebras to $\widehat{gl}(\infty ,{R}_{m})$ with ${R}_{m}=\u2102\left[T\right]/\left({T}^{m}\right)$ the truncated polynomial algebra is used. How $m$ has to be chosen depends on the weight $\lambda $. The classification of such representations which are unitary is given.

In the case of difference operators the representations of the more general algebra of pseudo-difference operators are studied. Similar results as above are obtained. Note that the algebras studied by the authors occur in the physicists’ literature as ${W}_{1+\infty}$- algebras.

##### MSC:

17B68 | Virasoro and related algebras |

17B10 | Representations of Lie algebras, algebraic theory |

17B70 | Graded Lie (super)algebras |

81T40 | Two-dimensional field theories, conformal field theories, etc. |

##### Keywords:

quasifinite highest weight modules; algebra of differential operators; central extension; highest weight modules; parabolic subalgebras; Verma modules; difference operators; representations; ${W}_{1+\infty}$- algebras##### References:

[1] | [F] Feigin, B.L.: The Lie algebrasgl($\lambda $) and the cohomology of the Lie algebra of differential operators. Usp. Math. Nauk35, No. 2, 157–158 (1988) |

[2] | [FL] Luk’yanov, S.L., Fateev, V.A.: Conformally invariant models of two-dimensional quantum field theory withZ n symmetry. Zh. Esp. Theor. Fiz.94, No. 3, 23–37 (1988) |

[3] | [FFZ] Fairlie, D., Fletcher, P., Zachos, C.: Phys. Lett.218B, 203 (1989) |

[4] | [GL] Golenishcheva-Kutuzova, M., Lebedev, D.: Vertex operator representation of some quantum tori Lie algebras. Commun. Math. Phys.148, 403–416 (1992) · Zbl 0766.17021 · doi:10.1007/BF02100868 |

[5] | [K] Kac, V.G.: Infinite-dimensional Lie algebras. 3-d edition, Cambridge: Cambridge University Press, 1990 |

[6] | [KP] Kac, V.G., Peterson, D.H.: Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981) · Zbl 0469.22016 · doi:10.1073/pnas.78.6.3308 |

[7] | [KhZ] Khesin, B., Zakharevich, I.: Poisson-Lie group of pseudodifferential symbols and fractional KP-KdV hiearchies. C.R. Acad. Sci. Paris,t. 316, Serie 1, 621–626 (1993) |

[8] | [Li] Li, W.L.: 2-cocycles on the algebra of differential operators. J. Algebra122, 64–80 (1989) · Zbl 0671.17010 · doi:10.1016/0021-8693(89)90237-8 |

[9] | [PSR] Pope, C.N., Shen, X., Romans, L.J.:W and the Racah-Wigner Algebra. Nucl. Phys.339B, 191–122 (1990) · doi:10.1016/0550-3213(90)90539-P |

[10] | [R] Radul, A.O.: Lie algebras of differential operators, their central extensions andW-algebras. Funct. Anal. Appl.25, No. 1, 86–91 (1991) · Zbl 0809.47044 · doi:10.1007/BF01090674 |

[11] | [RV] Radul, A.O., Vaysburd, I.: Differential operators andW-algebras. Phys. Lett.274B, 317–322 (1992) |

[12] | [Z] Zamolodchikov, A.B.: Infinite additional symmetries in 2-dimensional conformal quantum field theory. Teor. Mat. Phys.65, 1205–1213 (1985) · doi:10.1007/BF01036128 |