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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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