Algèbres 𝒲 et équations non-linéaires
Séminaire Bourbaki : volume 1997/98, exposés 835-849, Astérisque, no. 252 (1998), Exposé no. 839, 25 p.
@incollection{SB_1997-1998__40__105_0,
     author = {van Moerbeke, Pierre},
     title = {Alg\`ebres $\mathcal {W}$ et \'equations non-lin\'eaires},
     booktitle = {S\'eminaire Bourbaki : volume 1997/98, expos\'es 835-849},
     series = {Ast\'erisque},
     note = {talk:839},
     pages = {105--129},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {252},
     year = {1998},
     mrnumber = {1685581},
     zbl = {1029.81035},
     language = {fr},
     url = {http://archive.numdam.org/item/SB_1997-1998__40__105_0/}
}
TY  - CHAP
AU  - van Moerbeke, Pierre
TI  - Algèbres $\mathcal {W}$ et équations non-linéaires
BT  - Séminaire Bourbaki : volume 1997/98, exposés 835-849
AU  - Collectif
T3  - Astérisque
N1  - talk:839
PY  - 1998
SP  - 105
EP  - 129
IS  - 252
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/SB_1997-1998__40__105_0/
LA  - fr
ID  - SB_1997-1998__40__105_0
ER  - 
%0 Book Section
%A van Moerbeke, Pierre
%T Algèbres $\mathcal {W}$ et équations non-linéaires
%B Séminaire Bourbaki : volume 1997/98, exposés 835-849
%A Collectif
%S Astérisque
%Z talk:839
%D 1998
%P 105-129
%N 252
%I Société mathématique de France
%U http://archive.numdam.org/item/SB_1997-1998__40__105_0/
%G fr
%F SB_1997-1998__40__105_0
van Moerbeke, Pierre. Algèbres $\mathcal {W}$ et équations non-linéaires, dans Séminaire Bourbaki : volume 1997/98, exposés 835-849, Astérisque, no. 252 (1998), Exposé no. 839, 25 p. http://archive.numdam.org/item/SB_1997-1998__40__105_0/

[1] M. Adler : On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV equations, Inv. Math. 50, 219-248 (1979). | DOI | EuDML | Zbl

[2] M. Adler, P. Van Moerbeke: Completely integrable systems, Euclidean Lie algebras and Curves, Adv. Math. 38, 267-317 (1980). | DOI | Zbl

[3] M. Adler, P. Van Moerbeke: Birkhoff strata, Bäcklund transformations and regularization of isospectral operators, Adv. Math. 108, 140-204 (1994). | DOI | Zbl

[4] M. Adler, P. Van Moerbeke: A matrix integral solution to two-dimensional Wp- Gravity, Comm. Math. Phys. 147, 25-56 (1992). | DOI | Zbl

[5] M. Adler, P. Van Moerbeke : Compatible Poisson structures and the Virasoro algebra, Comm. Pure and Appl. Math. 47, 5-37 (1994). | DOI | Zbl

[6] M. Adler, A. Morozov, T. Shiota and P. Van Moerbeke : A matrix integral solution to [P,Q] = P and matrix Laplace transforms, Comm. Math. Phys. 180, 233-263 (1996). | DOI | Zbl

[7] M. Adler, T. Shiota and P. Van Moerbeke : A Lax pair representation for the vertex operator and the central extension, Comm. Math. Phys. 171, 547-588 (1995). | DOI | Zbl

[8] M. Adler, T. Shiota and P. Van Moerbeke : From the w∞-algebra to its central extension: a τ-function approach, Physics Letters A194, 33-43 (1994). | DOI | Zbl

[9] M. Adler, T. Shiota and P. Van Moerbeke : Random matrices, vertex operators and the Virasoro algebra, Phys. Lett. A208, 67-78 (1995). | DOI | Zbl

[10] M. Adler, T. Shiota and P. Van Moerbeke : Random matrices, Virasoro algebras and non-commutative KP, Duke math. J. 94, 379-431 (1998). | DOI | Zbl

[11] B. Bakalov, E. Horozov and M. Yakimov : General methods for constructing bispectral operators , Phys. Letters A, 222 59-66 (1996). | DOI | Zbl

[12] Bessis, D., Itzykson, Cl., Zuber, J.-B. : Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1, 109-157 (1980). | DOI | Zbl

[13] P. Bouwknegt, K. Schoutens: W-symmetry in conformal field theory, Phys. Rep. 223, 183-286 (1993). | DOI | Zbl

[14] E. Date, M. Jimbo, M. Kashiwara, T. Miwa: Transformation groups for soliton equations, Proc. RIMS Symp. Nonlinear integrable systems, Classical and quantum theory (Kyoto 1981), pp. 39-119. Singapore: World Scientific (1983). | Zbl

[15] L. Dickey: Soliton equations and integrable systems, World Scientific (1991). | Zbl

[16] L. Dickey : Additional symmetries of KP, Grassmannian, and the string equation, preprint 1992. | Zbl

[17] L. Dickey: Lectures on classical W-algebras (Cortona Lectures), Acta Appl. Math. 47, 243-321 (1997). | DOI | Zbl

[18] P. Di Francesco, Ci. Itzykson, J.-B. Zuber: Classical W-algebras, Comm. Math. Phys. 140, 543-567 (1991). | DOI | Zbl

[19] F. Dyson : Fredholm determinants and inverse scattering problems, Commun. Math. Phys. 47, 171-183 (1976). | DOI | Zbl

[20] F. Dyson: Statistical theory of energy levels of complex systems, I, II and III, J. Math Phys 3 140-156, 157-165, 166-175 (1962). | DOI | Zbl

[21] J. Fastré: Boson-correspondence for W-algebras, Bäcklund-Darboux transformations and the equation [L, P] = Ln, Bull. des Sciences Math. (1997)

[22] V. A. Fateev, S. L. Lukyanov : Additional symmetries and exactly solvable models of two-dimensional conformal field theory, Int. J. Modern Phys. A3 507 (1988). | DOI

[23] L. Fehér, L. O.'Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf: On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. reports 222(1),1-64 (1992). | DOI

[24] E. Frenkel, V. Kac, A. Radul, W. Wang:W1+∞ and WglN with central charge N, Comm. Math. Phys. 170, 337-357 (1995). | DOI | Zbl

[25] E. Frenkel and N. Reshetikin: Quantum affine algebras and deformations of the Virasoro and W-algebras, Comm. Math. Physics (1997). | Zbl

[26] I. M. Gardner: KdV equation and generalizations IV, J. Math. Phys. 12, 1548-1551 (1971). | Zbl

[27] I. M. Gel'Fand, L. Dickey: Family of Hamiltonian structures connected with integrable non-linear differential equations, Funct. Anal. Appl. 2, 92-93 (1968).

[28] I. M. Gel'Fand, D. B. Fuks: Cohomologies of the Lie algebra of vector fields on the circle, Funct. Anal. Appl. 2, 92-93 (1968). | Zbl

[29] J.-L. Gervais: Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson bracket, Phys. letters 16013, 277 (1985).

[30] L. Haine, E. Horozov : Toda Orbits of Laguerre Polynomials and Representations of the Virasoro Algebra, Bulletin des Sciences Math. (1993). | Zbl

[31] M. Jimbo, T. Miwa, Y. Mori and M. Sato: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica 1D, 80-158 (1980). | Zbl

[32] V. Kac, A. Radul: Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157, 429-457 (1993). | DOI | Zbl

[33] V. G. Kac, A. K. Raina: Bombay lectures on highest weight representations of infinite dimensional Lie Algebras, Adv. Series Math. Phys. vol. 2, 1987. | Zbl

[34] V. Kac, A. Schwarz: Geometric interpretation of partition function of 2D-gravity, Phys. lett. 257B, 329-334 (1991). | DOI

[35] A. A. Kirillov: Orbits of the group of diffeomorphisms of a circle and local Lie superalgebras, Funkt. Anal. Appl. 21, 19-55 (1981). | Zbl

[36] M. Kontsevich: Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147, 1-23 (1992). | DOI | Zbl

[37] F. Magri: A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19, 1156-1162 (1978). | DOI | Zbl

[38] M. L. Mehta: Random matrices, 2nd ed. Boston: Acad. Press, 1991. | Zbl

[39] A. Y. Orlov, E. I. Schulman: Additional Symmetries for Integrable and Conformal Algebra Representation, Letters in Math. Phys. 12, 171-179 (1986). | DOI | Zbl

[40] C. E. Porter and N. Rosenzweig, Statistical properties of atomic and nuclear spectra, Ann. Acad. Sci. Fennicae, Serie A, VI Physica 44, 1-66 (1960); Repulsion of energy levels in complex atomic spectra, Phys. Rev 120, 1698-1714 (1960). | Zbl

[41] A. Radul: Lie algebras of differential operators, their central extensions, and W-algebras, Funct. Anal. Appl. 25, 33-49 (1991). | DOI | Zbl

[42] P. Sarnak: Arithmetic quantum chaos, Israel Math. Conf. Proceedings, 8, 183-236 (1995). | Zbl

[43] K. Takasaki, T. Takebe : Integrable hierarchies and dispersionless limit, Reviews in Math. Phys. 7, 743-808 (1995) . | DOI | Zbl

[44] C. A. Tracy, H. Widom: Level-spacings distribution and the Airy kernel, Comm. Math. Phys. 159, 151-174 (1984). | DOI | Zbl

[45] K. Ueno, K. Takasaki : Toda Lattice Hierarchy, Adv. Studies in Pure Math. 4, 1-95 (1984). | Zbl

[46] J. Van De Leur: The W1+∞(gls)-symmetries of the s-component KP hierarchy, J. of Math. Phys. 37, 2315-2337 (1996). | DOI | Zbl

[47] P. Van Moerbeke : Integrable foundations of string theory, in Lectures on Integrable systems, Proceedings of the CIMPA-school, 1991, Ed.: O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, World scientific, 163-267 (1994). | Zbl

[48] P. Van Moerbeke: The spectrum of random matrices and integrable systems, Group 21, Physical applications and Mathematical aspects of Geometry, Groups and Algebras, Vol.II, 835-852, Eds. : H.-D. Doebner, W. Scherer, C. Schulte, World scientific, Singapore, 1997.

[49] P. Van Moerbeke: Cours à l'Institut H. Poincaré (automne 1996).

[50] E. P. Wigner:On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambr. Phil. Soc. 47, 790-798 (1951). | DOI | Zbl

[51] Witten, Ed. : Two-dimensional gravity and intersection theory of moduli space, Harvard University lecture, May 1990, Journal of diff. geometry, 1991. | Zbl

[52] V. Zakharov, L. D. Faddeev: The KdV equation is a completely integrable Hamiltonian system, Funct. Anal. Appl. 5, 18-27 (1971). | Zbl

[53] A. B. Zamolodchikov: Infinite additional symmetries in two-dimensional conformal quantum field theory, Theoret. Math. Phys. 65, 1205 (1985). | DOI