Projective unitary positive-energy representations of \(Diff(S^ 1)\)
type: Item
instance of: scholarly article
title: Projective unitary positive-energy representations of \(Diff(S^ 1)\)
zbMATH Open document ID: 0636.22013
DOI: 10.1016/0022-1236(85)90090-4
author: Roe Goodman
author: Nolan Wallach
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: 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 \({\mathfrak h}\subset \hat {\mathfrak d}\). \textit{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.
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 58D05
Mathematics Subject Classification ID: 58B25
Mathematics Subject Classification ID: 58H05
zbMATH DE Number: 4036098
zbMATH Keywords: contravariant form
zbMATH Keywords: Lie algebra of complex vector fields
zbMATH Keywords: Fourier polynomials
zbMATH Keywords: Virasoro algebra
zbMATH Keywords: triangular decomposition
zbMATH Keywords: highest weight modules
zbMATH Keywords: continuous projective representation
zbMATH Keywords: orientation preserving diffeomorphisms
reviewed by: Jiří Vanžura
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90090-4
OpenAlex ID: W2089803146
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description: scientific article; zbMATH DE number 4036098