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Zbl 0826.17027
Quasifinite highest weight modules over the Lie algebra of differential operators on the circle.
(English)
[J] Commun. Math. Phys. 157, No.3, 429-457 (1993). ISSN 0010-3616; ISSN 1432-0916/e

In the article it is shown how the algebra of differential operators $\cal D$ on the circle, respectively its universal central extension $\widehat {\cal D}$, can be obtained via a twisted Laurent polynomial algebra over a polynomial algebra. In a similar way the algebras ${\cal D}_q$ 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 $M(\lambda)$ associated to the weight $\lambda$ one obtains the unique irreducible highest weight module $L(\lambda)$ as a quotient.\par 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 $L(\lambda)$ is quasifinite if and only if $L(\lambda)$ is a quotient of a generalized Verma module $M(\lambda, {\germ p})$ with respect to a parabolic subalgebra $\germ p$ (or equivalently to a characteristic polynomial). The authors give necessary and sufficient conditions for $L(\lambda)$ 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 $R_m = \bbfC[T] / (T^m)$ 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.\par 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.
[M.Schlichenmaier (Mannheim)]
MSC 2000:
*17B68 Virasoro and related algebras
17B10 Representations of Lie algebras, algebraic theory
81T40 Two-dimensional field theories, etc.

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

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