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Projective unitary positive-energy representations of \(\operatorname{Diff}(S^1)\). (English) Zbl 0636.22013

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.

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

Citations:

Zbl 0493.17011
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References:

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