Goodman, Roe; Wallach, Nolan R. Projective unitary positive-energy representations of \(\operatorname{Diff}(S^1)\). (English) Zbl 0636.22013 J. Funct. Anal. 63, 299-321 (1985). Let \({\mathfrak d}\) be the Lie algebra of complex vector fields on \(S^1\) whose coefficients are the Fourier polynomials. Its central extension – the Virasoro algebra \(\hat{\mathfrak d}\) – admits a triangular decomposition \(\hat{\mathfrak d}={\mathfrak n}^+\oplus {\mathfrak h}\oplus {\mathfrak n}^-\), which enables us to construct highest weight modules \(L(h,c)\) for \(\hat{\mathfrak d}\) depending on two complex parameters \(h\) and \(c\) determined by the action of the subalgebra \({\mathfrak h}\subset \hat {\mathfrak d}\). V. G. Kac conjectured [Lect. Notes Math. 933, 117–126 (1982; Zbl 0493.17011)] that if the contravariant \((=\) Hermitian Shapavolov) form on \(L(h,c)\) is positive definite, then the module \(L(h,c)\) “integrates” to a continuous projective representation of the group \({\mathcal D}\) of orientation preserving diffeomorphisms of \(S^1\). The authors prove the Kac conjecture. Let us remark that the arguments in the proof are rather delicate. Reviewer: Jiří Vanžura (Brno) Cited in 43 Documents MSC: 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 17B66 Lie algebras of vector fields and related (super) algebras 17B68 Virasoro and related algebras 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 58B25 Group structures and generalizations on infinite-dimensional manifolds 58H05 Pseudogroups and differentiable groupoids Keywords:contravariant form; Lie algebra of complex vector fields; Fourier polynomials; Virasoro algebra; triangular decomposition; highest weight modules; continuous projective representation; orientation preserving diffeomorphisms Citations:Zbl 0493.17011 PDFBibTeX XMLCite \textit{R. Goodman} and \textit{N. R. Wallach}, J. Funct. Anal. 63, 299--321 (1985; Zbl 0636.22013) Full Text: DOI References: [1] Chodos, A.; Thorn, C., Making the massless string massive, Nuclear Phys. B, 72, 509-522 (1974) [2] Goodman, R.; Wallach, N. R., Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. Reine Angew. Math., 347, 69-133 (1984) · Zbl 0514.22012 [3] Hamilton, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. N. S., 7, 65-222 (1982) · Zbl 0499.58003 [4] Kac, V. G., Highest weight representations of infinite dimensional Lie algebras, (Proceedings of ICM. Proceedings of ICM, Helsinki (1978)), 299-304 · Zbl 0425.17009 [5] Kac, V. G., Some problems on infinite-dimensional Lie algebras and their representations, (Lie Algebras and Related Topics. Lie Algebras and Related Topics, Lecture Notes in Mathematics, Vol. 933 (1982), Springer: Springer Berlin/Heidelberg/New York) · Zbl 0425.17009 [6] Nelson, E., Time-ordered operator products of sharp-time quadratic forms, J. Funct. Anal., 11, 211-219 (1972) · Zbl 0239.47012 [7] Neretin, Yu. A., Funct. Anal. Appl., 17, 235-236 (1983), Russian; English trans. · Zbl 0574.43005 [8] Reed, M.; Simon, B., Methods of Modern Mathematical Physics II, (Fourier Analysis, Self-Adjointness (1975), Academic Press: Academic Press New York) · Zbl 0517.47006 [9] Segal, G., Unitary representations of some infinite dimensional groups, Comm. Math. Phys., 80, 301-342 (1981) · Zbl 0495.22017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.