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Spinor representation of the infinite-dimensional orthogonal semigroup and the Virasoro algebra. (Russian) Zbl 0693.17010

The author considers a category of pairs of Hilbert spaces \((V_+,V_- )\) with a fixed antilinear isometry mapping \(V_-\) onto \(V_+\) and morphisms defined by block matrices with Hilbert-Schmidt operators in the diagonal. A projective representation Spin by operators on a Fermion Fock space \(\Lambda (V_+)\) endowed with a family of seminorms making it a Fréchet space generated by the Grassmann products of basis elements of \(V_+\) is defined and studied on various subcategories. It is proved, that an irreducible highest weight representation of the Virasoro algebra is integrable to a projective representation of the group \(\mathrm{Diff}\) of analytic orientation preserving diffeomorphisms of the circle.
This representation extends to a projective representation of a semigroup, which was defined by the author as the complexification of \(\mathrm{Diff}\) [compare the author’s paper in ibid. 21, No. 2, 82–83 (1987; Zbl 0628.58004)]. Moreover it is stated, that every irreducible highest weight representation of the Virasoro algebra admits a realization by sections in a holomorphic linear bundle on a Siegel-Kirillov domain.

MSC:

17B68 Virasoro and related algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds

Citations:

Zbl 0628.58004
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