Geodesic
Holomorphic extensions of representations of the group of the diffeomorphisms of the circle
Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 75-97 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_1990_67_1_a4,
     author = {Yu. A. Neretin},
     title = {Holomorphic extensions of representations of the~group of the~diffeomorphisms of the~circle},
     journal = {Sbornik. Mathematics},
     pages = {75--97},
     year = {1990},
     volume = {67},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1990_67_1_a4/}
}
TY  - JOUR
AU  - Yu. A. Neretin
TI  - Holomorphic extensions of representations of the group of the diffeomorphisms of the circle
JO  - Sbornik. Mathematics
PY  - 1990
SP  - 75
EP  - 97
VL  - 67
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1990_67_1_a4/
LA  - en
ID  - SM_1990_67_1_a4
ER  - 
%0 Journal Article
%A Yu. A. Neretin
%T Holomorphic extensions of representations of the group of the diffeomorphisms of the circle
%J Sbornik. Mathematics
%D 1990
%P 75-97
%V 67
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1990_67_1_a4/
%G en
%F SM_1990_67_1_a4
Yu. A. Neretin. Holomorphic extensions of representations of the group of the diffeomorphisms of the circle. Sbornik. Mathematics, Tome 67 (1990) no. 1, pp. 75-97. http://geodesic.mathdoc.fr/item/SM_1990_67_1_a4/

[1] Azizov T. Ya., Iokhvidov I. S., Osnovy teorii lineinykh operatorov v prostranstvakh s indefinitnoi metrikoi, Nauka, M., 1986 | MR

[2] Berezin F. A., Metod vtorichnogo kvantovaniya, Nauka, M., 1965 | MR

[3] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[4] Kirillov A. A., “Kelerova struktura na $K$-orbitakh gruppy diffeomorfizmov okruzhnosti”, Funktsion. analiz i ego pril., 21:2 (1987), 2–45 | MR | Zbl

[5] Kirillov A. A., Yurev D. V., “Kelerova geometriya prostranstva $M=\operatorname{Diff}_+(S^1)/\operatorname{Rot}(S^1)$”, Fynktsion. analiz i ego pril., 21:4 (1987), 35–46 | MR | Zbl

[6] Kontsevich M. L., “Algebra Virasoro i prostranstva Teikhmyullera”, Funktsion. analiz i ego pril., 21:2 (1987), 78–79 | MR | Zbl

[7] Krichever I. M., Novikov S. P., “Algebry tipa Virasoro, rimanovy poverkhnosti i struktury teorii solitonov”, Funktsion. analiz i ego pril., 21:2 (1987), 46–63 | MR | Zbl

[8] Miranda K., Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, M., 1957

[9] Neretin Yu. A., “Dopolnitelnaya seriya predstavlenii gruppy diffeomorfizmov okruzhnosti”, UMN, 37:2 (1982), 213–214 | MR | Zbl

[10] Neretin Yu. A., “Unitarnye predstavleniya gruppy diffeomorfizmov okruzhnosti so starshim vesom”, Funktsion. analiz i ego pril., 17:3 (1983), 85–86 | MR | Zbl

[11] Neretin Yu. A., Unitarnye predstavleniya algebry Virasoro so starshim vesom, Dis. ... kand. fiz.-matem. nauk, MGU, M., 1983

[12] Neretin Yu. A., “O spinornom predstavlenii $O(\infty,\mathbf{C})$”, DAN SSSR, 289:2 (1986), 282–285 | MR

[13] Neretin Yu. A., “O kompleksnoi polugruppe, soderzhaschei gruppu diffeomorfizmov okruzhnosti”, Funktsion. analiz i ego pril., 21:2 (1987), 82–83 | MR | Zbl

[14] Neretin Yu. A., “Predstavleniya algebry Virasoro i affinnykh algebr”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 21, VINITI, M., 1988, 163–224 | MR

[15] Olshanskii G. I., “Invariantnye konusy v gruppakh Li, polugruppy Li i golomorfnaya diskretnaya seriya”, Funktsion. analiz i ego pril., 15:4 (1984), 58–66 | MR

[16] Duren P. L., Univalent functions, Springer, N. Y., 1983 | MR | Zbl

[17] Goddard P., Kent A., “Olive D. Unitary representations of Virasoro and super Virasoro algebra”, Comm. Math. Phys., 103:1 (1986), 105–119 | DOI | MR | Zbl

[18] Friedan D., Qiu Z., Shenker S., “Conformal invariance, unitarity and two-dimensional critical exponents”, Vertex operators in mathematics and physics, Springer,, N. Y., 1984, 419–450 | MR

[19] Goodman R., Wallach N. R., “Projective unitary positive-energy representations of $\operatorname{Diff}(S^1)$”, J. Funct. Anal., 63:3 (1985), 299–321 | DOI | MR | Zbl

[20] Pressley A., Segal G., Loop groups, Clarendon Press, Oxford, 1986 | MR | Zbl

[21] Segal G., “Unitary representations of some infinite-dimensional groups”, Comm. Math. Phys., 80:3 (1981), 301–342 | DOI | MR | Zbl

[22] Souriau J. M., “Sur les groups diffeologiques”, Lect. Notes in Math., 836 (1980), 91–128 | DOI | MR | Zbl

[23] Witten E., “Quantum field theory, grassmanians and algebraic curves”, Comm. Math. Phys., 113:4 (1988), 1–54 | DOI | MR

[24] Lebedev N. A., Printsip ploschadei v teorii odnolistnykh funktsii, Nauka, M., 1975 | MR | Zbl

[25] Krichever I. M., “Integrirovanie nelineinykh uravnenii metodami algebraicheskoi geometrii”, Funktsion. analiz i ego pril., 11:1 (1977), 15–31 | MR | Zbl