Quasifinite Representations of Classical Subalgebras of the Lie Superalgebra of Quantum Pseudodifferential Operators

We classify the anti-involutions of the superalgebra of quantum pseudodifferential operators on the super circle preserving the principal gradation, producing in this way a family of Lie subalgebras minus fixed by these anti-involutions. We classify the irreducible quasifinite highest weight representations of the central extension of these Lie subalgebras. 1. Introduction The -infinity algebras naturally arise in various physical theories, such as conformal field theory and the theory of quantum Hall effect (see [1, 2] and references therein). The algebra, which is the central extension of the Lie algebra of differential operators on the circle, is the most fundamental among these algebras. The difficulty in understanding the representation theory of a Lie algebra of this kind is that although admits a natural -gradation, each of the graded subspaces is still infinite dimensional in contrast to the more familiar cases such as the Virasoro algebra and Kac-Moody algebras. Therefore, the study of the highest weight modules which satisfy the quasifiniteness condition, that its graded subspaces have finite dimension, becomes a nontrivial problem. The systematic study of quasifinite highest weight modules of was initiated by Kac and Radul in [2] and further studied in [1, 3–5] and many others. By analyzing for which parabolic subalgebras of？？ the corresponding generalized Verma modules are quasifinite, Kac and Radul [2] gave a characterization of quasifinite highest weight -modules in terms of certain generating function of highest weights and these modules where constructed in terms of irreducible highest weight representations of the Lie algebra of infinite matrices. The classification and construction of quasifinite modules for the matrix version (denoted by ), super analog, -analog, and super -analog of , were developed in [1, 2, 4, 6], respectively. The Lie algebra , recently studied in [4], correspond to the central extension of the algebra of matrix differential operators on the circle. The study of the representation theory of some interesting subalgebras of , and its -analog and super version [1], is not complete. Another important example is the Lie algebra which is a particular case of a family of subalgebras of , where ( ) is the central extension of the Lie algebra of differential operators on the circle that are a multiple of . This Lie algebra was studied by Kac and Liberati in [3]; observe that . Following the ideas of Kac-Radul [2], in [3] they obtained the classification of the irreducible quasifinite highest weight modules over . They also