We classify anti-involutions of Lie superalgebra SD preserving the principal gradation, where 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 _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.
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Shun-Jen Cheng, Weiqiang Wang, Lie Subalgebras of Differential Operators on the Super Circle. Publ. Res. Inst. Math. Sci. 39 (2003), no. 3, pp. 545–600DOI 10.2977/PRIMS/1145476079