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 Mat. Sb., 1989, Volume 180, Number 5, Pages 635–657 (Mi msb1625)

Holomorphic extensions of representations of the group of the diffeomorphisms of the circle

Yu. A. Neretin

Abstract: This paper gives the construction of a semigroup $\Gamma$ which could be thought of as the complexincation of the group $\operatorname{Diff}$ of analytic diffeomorphisms of the circle, and it is shown that any unitary projective representation of $\operatorname{Diff}$ with highest weight has a holomorphic extension to $\Gamma$. For this, $\Gamma$ is embedded in the semigroup of “endomorphisms of canonical commutation relations” (this is a certain part of the Lagrange Grassmannian in complex symplectic Hilbert space).
Bibliography: 25 titles.

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English version:
Mathematics of the USSR-Sbornik, 1990, 67:1, 75–97

Bibliographic databases:

UDC: 517.9
MSC: 22E65, 17B65

Citation: Yu. A. Neretin, “Holomorphic extensions of representations of the group of the diffeomorphisms of the circle”, Mat. Sb., 180:5 (1989), 635–657; Math. USSR-Sb., 67:1 (1990), 75–97

Citation in format AMSBIB
\Bibitem{Ner89} \by Yu.~A.~Neretin \paper Holomorphic extensions of representations of the~group of the~diffeomorphisms of the~circle \jour Mat. Sb. \yr 1989 \vol 180 \issue 5 \pages 635--657 \mathnet{http://mi.mathnet.ru/msb1625} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1007467} \zmath{https://zbmath.org/?q=an:0696.58010} \transl \jour Math. USSR-Sb. \yr 1990 \vol 67 \issue 1 \pages 75--97 \crossref{https://doi.org/10.1070/SM1990v067n01ABEH001321} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1990ED88000005} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. A. Neretin, “A semigroup of operators in the boson fock space”, Funct. Anal. Appl., 24:2 (1990), 135–144
2. Neretin I., “On Operators Connecting Holomorphic Representations of Different Groups”, 312, no. 6, 1990, 1318–1321
3. Yu. A. Neretin, “Categories of bistochastic measures, and representations of some infinite-dimensional groups”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 197–219
4. Yu. A. Neretin, “Universal completions of complex classical groups”, Funct. Anal. Appl., 26:4 (1992), 254–265
5. D. V. Yur'ev, “Quantum projective field theory: Quantum-field analogs of the Euler–Arnol'd equations in projective $G$ multiplets”, Theoret. and Math. Phys., 98:2 (1994), 147–161
6. B. Julia, H. Nicolai, “Conformal internal symmetry of 2d σ-models coupled to gravity and a dilaton”, Nuclear Physics B, 482:1-2 (1996), 431
7. Yu. A. Neretin, “On the correspondence between boson Fock space and the $L^2$ space with respect to Poisson measure”, Sb. Math., 188:11 (1997), 1587–1616
8. Yu. A. Neretin, “Supercomplete Bases in the Space of Symmetric Functions”, Funct. Anal. Appl., 32:1 (1998), 10–22
9. Helene Airault, “Affine coordinates and Virasoro unitarizing measures”, Journal de Mathématiques Pures et Appliquées, 82:4 (2003), 425
10. Helene Airault, Vladimir Bogachev, “Realization of Virasoro unitarizing measures on the set of Jordan curves”, Comptes Rendus Mathematique, 336:5 (2003), 429
11. Yu. A. Neretin, “$K$-Finite Matrix Elements of Irreducible Harish-Chandra Modules are Hypergeometric”, Funct. Anal. Appl., 41:4 (2007), 295–302
12. Richard Blute, Prakash Panangaden, Dorette Pronk, “Conformal Field Theory as a Nuclear Functor”, Electronic Notes in Theoretical Computer Science, 172 (2007), 101
13. Helene Airault, Yuri A. Neretin, “On the action of Virasoro algebra on the space of univalent functions”, Bulletin des Sciences Mathématiques, 132:1 (2008), 27
14. Theodore Erler, Martin Schnabl, “A simple analytic solution for Tachyon condensation”, J. High Energy Phys, 2009:10 (2009), 066
15. Neretin Yu.A., “Ramified Integrals, Casselman Phenomenon, and Holomorphic Continuations of Group Representations”, Analysis and Mathematical Physics, Trends in Mathematics, eds. Gustafsson B., Vasilev A., Birkhauser Boston, 2009, 427–439
16. KARL-HERMANN NEEB, “SEMIBOUNDED REPRESENTATIONS AND INVARIANT CONES IN INFINITE DIMENSIONAL Lie ALGEBRAS”, Confluentes Math, 02:01 (2010), 37
17. Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type”, Russian Math. Surveys, 70:4 (2015), 715–773
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