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Representations of the Virasoro algebra and affine algebras. (Russian) Zbl 0656.17011
The author gives a survey of basic properties and constructions of representations of the Virasoro algebra and affine algebras as well as infinite-dimensional Lie groups connected with them. 28 references up to 1984 are listed plus the author’s 1986 Doklady paper.
The chapter headings are as follows:
(1) Diffeomorphism groups of the circle. (2) Verma modules over Virasoro algebras (a.o. Shapovalov form). (3) Second quantization method (a.o. bosonic and fermionic Fock spaces; spinor representations; Weil representation; representations of (G,K)-pairs; Araki scheme). (4) Almost invariant structures. (5) Unitary representations of Diff. (6) Affine algebra (a.o. root systems; Dynkin diagram). (7) Representations of the groups \(Diff\ltimes C^{\infty}(S^ 1,K)\) (a.o. Goddard-Kent-Olive construction). (8) Constructions of basic modules (a.o. fermionic constructions; Segal constructions for \(s\ell_ 2)\). (9) Holomorphic extensions (a.o. partial complexification of the group Diff; representations of the complex group \(C^{\infty}(S^ 1,K)\); Siegel domain for \(Diff_ a)\).

MSC:
17B65 Infinite-dimensional Lie (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
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