Lie subalgebras of differential operators on the super circle

Details Lie subalgebras of differential operators on the super circle Lie subalgebras of differential operators on the super circle

2001-03-15

arxiv.org/abs/math/0103092v1

Publ. Res. Inst. Math. Sci. 39 (2003), No. 3, 545--600

Mathematics

Quantum Algebra

51 pages, Latex format

Scientific paper

We classify anti-involutions of Lie superalgebra $\hsd$ preserving the principal gradation, where $\hsd$ 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 $\hat{gl}_{\infty|\infty}$ 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.

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