Lie Subalgebras of Differential Operators on the Super Circle

Abstract

We classify anti-involutions of Lie superalgebra $\widehat{\mathcal{S}\mathcal{D}}$ preserving the principal gradation, where $\widehat{\mathcal{S}\mathcal{D}}$ is the central extension of the Lie superalgebra of differential operators on the super circle $S^{1\mid 1}$. We clarify the relations between the corresponding subalgebras fixed by these anti-involutions and subalgebras of ${\widehat{gl}}_{\infty |\infty }$ 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.