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Representations of the Virasoro algebra. (English) Zbl 0722.17020

Representation of Lie groups and related topics, Adv. Stud. Contemp. Math. 7, 465-554 (1990).
[For the entire collection see Zbl 0714.00007.]
This most comprehensive treatise on the representation theory of the famous Virasoro algebra was preprinted 10 years ago, but apart from a version (marred by funny English) preprinted in D. Leites (ed.), Seminar on supermanifolds No.4, Reports of Stockholm University (1987), this important work was unavailable.
The paper contains a detailed description of Verma modules and the modules of semi-infinite forms (in particular, the space of semi-infinite forms is endowed with \(\mathfrak{gl}(\infty)\)-, \(\mathfrak{sp}(2\infty)\) or an \(\mathfrak o(\infty)\)-module structure). The intertwining operators between modules of semi-infinite forms over the Virasoro algebra are identified as “vertex operators” of physicists. The vertex operators are studied in detail: they are instrumental in the proofs of the main theorems and elsewhere. Proofs are lucid.

MSC:

17B68 Virasoro and related algebras
17B69 Vertex operators; vertex operator algebras and related structures
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Citations:

Zbl 0714.00007