Lie Subalgebras of Differential Operators on the Super Circle

Abstract

We classify anti-involutions of Lie superalgebra \widehat{\mathcal{SD}} preserving the principal gradation, where \widehat{\mathcal{SD}} is the central extension of the Lie superalgebra of differential operators on the super circle S^1|1. We clarify the relations between the corresponding subalgebras fixed by these anti-involutions and subalgebras of \widehat{gl}_∞|∞ of types OSP and P. We obtain a criterion for an irreducible highest weight module over these subalgebras to be quasifinite and construct free field realizations of a distinguished class of these modules. We further establish dualities between them and certain finite-dimensional classical Lie groups on Fock spaces.