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Quasifinite highest weight modules over the Lie algebra of differential operators on the circle. (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 $$M(\lambda)$$ associated to the weight $$\lambda$$ one obtains the unique irreducible highest weight module $$L(\lambda)$$ 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 $$L(\lambda)$$ is quasifinite if and only if $$L(\lambda)$$ is a quotient of a generalized Verma module $$M(\lambda, {\mathfrak p})$$ with respect to a parabolic subalgebra $$\mathfrak p$$ (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 $$R_m = \mathbb{C}[T] / (T^m)$$ 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.
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 and Lie superalgebras, algebraic theory (weights) 17B70 Graded Lie (super)algebras 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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