Algèbres et équations non-linéaires
Séminaire Bourbaki (1997-1998)
- Volume: 40, page 105-129
- ISSN: 0303-1179
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topvan Moerbeke, Pierre. "Algèbres $\mathcal {W}$ et équations non-linéaires." Séminaire Bourbaki 40 (1997-1998): 105-129. <http://eudml.org/doc/110241>.
@article{vanMoerbeke1997-1998,
author = {van Moerbeke, Pierre},
journal = {Séminaire Bourbaki},
keywords = {-algebras; nonlinear equations; second symplectic structure; Hamiltonian vectors fields},
language = {fre},
pages = {105-129},
publisher = {Société Mathématique de France},
title = {Algèbres $\mathcal \{W\}$ et équations non-linéaires},
url = {http://eudml.org/doc/110241},
volume = {40},
year = {1997-1998},
}
TY - JOUR
AU - van Moerbeke, Pierre
TI - Algèbres $\mathcal {W}$ et équations non-linéaires
JO - Séminaire Bourbaki
PY - 1997-1998
PB - Société Mathématique de France
VL - 40
SP - 105
EP - 129
LA - fre
KW - -algebras; nonlinear equations; second symplectic structure; Hamiltonian vectors fields
UR - http://eudml.org/doc/110241
ER -
References
top- [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). Zbl0393.35058
- [2] M. Adler, P. van Moerbeke: Completely integrable systems, Euclidean Lie algebras and Curves, Adv. Math.38, 267-317 (1980). Zbl0455.58017
- [3] M. Adler, P. van Moerbeke: Birkhoff strata, Bäcklund transformations and regularization of isospectral operators, Adv. Math.108, 140-204 (1994). Zbl0814.35114
- [4] M. Adler, P. van Moerbeke: A matrix integral solution to two-dimensional Wp- Gravity, Comm. Math. Phys.147, 25-56 (1992). Zbl0756.35074
- [5] M. Adler, P. van Moerbeke : Compatible Poisson structures and the Virasoro algebra, Comm. Pure and Appl. Math.47, 5-37 (1994). Zbl0801.58014
- [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). Zbl0858.35109
- [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). Zbl0839.35116
- [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). Zbl0961.37514
- [9] M. Adler, T. Shiota and P. van Moerbeke : Random matrices, vertex operators and the Virasoro algebra, Phys. Lett. A208, 67-78 (1995). Zbl1020.82577
- [10] M. Adler, T. Shiota and P. van Moerbeke : Random matrices, Virasoro algebras and non-commutative KP, Duke math. J.94, 379-431 (1998). Zbl1061.37047
- [11] B. Bakalov, E. Horozov and M. Yakimov : General methods for constructing bispectral operators , Phys. LettersA, 22259-66 (1996). Zbl0972.37545
- [12] Bessis, D., Itzykson, Cl., Zuber, J.-B. : Quantum field theory techniques in graphical enumeration, Adv. Appl. Math.1, 109-157 (1980). Zbl0453.05035
- [13] P. Bouwknegt, K. Schoutens: W-symmetry in conformal field theory, Phys. Rep.223, 183-286 (1993).
- [14] E. Date, M. Jimbo, M. Kashiwara, T. Miwa: Transformation groups for soliton equations, Proc. RIMS Symp. Nonlinear integrable systems, Classical and quantum theory (Kyoto1981), pp. 39-119. Singapore: World Scientific (1983). Zbl0571.35098
- [15] L. Dickey: Soliton equations and integrable systems, World Scientific (1991). Zbl0753.35075
- [16] L. Dickey : Additional symmetries of KP, Grassmannian, and the string equation, preprint 1992. Zbl1020.37555
- [17] L. Dickey: Lectures on classical W-algebras (Cortona Lectures), Acta Appl. Math.47, 243-321 (1997). Zbl0882.58024
- [18] P. Di Francesco, CI. Itzykson, J.-B. Zuber: Classical W-algebras, Comm. Math. Phys.140, 543-567 (1991). Zbl0752.17026
- [19] F. Dyson : Fredholm determinants and inverse scattering problems, Commun. Math. Phys.47, 171-183 (1976). Zbl0323.33008
- [20] F. Dyson: Statistical theory of energy levels of complex systems, I, II and III, J. Math Phys3140-156, 157-165, 166-175 (1962). Zbl0105.41604
- [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. A3507 (1988).
- [23] L. Fehér, L.O.' Raifeartaigh, P. Ruelle, I. Tsutsui, A. Wipf: On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. reports222(1),1-64 (1992).
- [24] E. Frenkel, V. Kac, A. Radul, W. Wang:W1+∞ and WglN with central charge N, Comm. Math. Phys.170, 337-357 (1995). Zbl0838.17028
- [25] E. Frenkel and N. Reshetikin: Quantum affine algebras and deformations of the Virasoro and W-algebras, Comm. Math. Physics (1997). Zbl0869.17014
- [26] I.M. Gardner: KdV equation and generalizations IV, J. Math. Phys.12, 1548-1551 (1971). Zbl0283.35021
- [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). Zbl0176.11501
- [29] J.-L. Gervais: Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson bracket, Phys. letters16013, 277 (1985).
- [30] L. Haine, E. Horozov : Toda Orbits of Laguerre Polynomials and Representations of the Virasoro Algebra, Bulletin des Sciences Math. (1993). Zbl0831.17011
- [31] M. Jimbo, T. Miwa, Y. Mori and M. Sato: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica1D, 80-158 (1980). Zbl1194.82007
- [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). Zbl0826.17027
- [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. Zbl0668.17012
- [34] V. Kac, A. Schwarz: Geometric interpretation of partition function of 2D-gravity, Phys. lett.257B, 329-334 (1991).
- [35] A.A. Kirillov: Orbits of the group of diffeomorphisms of a circle and local Lie superalgebras, Funkt. Anal. Appl.21, 19-55 (1981). Zbl0474.58003
- [36] M. Kontsevich: Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys.147, 1-23 (1992). Zbl0756.35081
- [37] F. Magri: A simple model of the integrable Hamiltonian equation, J. Math. Phys.19, 1156-1162 (1978). Zbl0383.35065
- [38] M.L. Mehta: Random matrices, 2nd ed. Boston: Acad. Press, 1991. Zbl0780.60014
- [39] A.Y. Orlov, E.I. Schulman: Additional Symmetries for Integrable and Conformal Algebra Representation, Letters in Math. Phys.12, 171-179 (1986). Zbl0618.35107
- [40] C.E. Porter and N. Rosenzweig, Statistical properties of atomic and nuclear spectra, Ann. Acad. Sci. Fennicae, Serie A, VIPhysica44, 1-66 (1960); Repulsion of energy levels in complex atomic spectra, Phys. Rev120, 1698-1714 (1960). Zbl0092.23304
- [41] A. Radul: Lie algebras of differential operators, their central extensions, and W-algebras, Funct. Anal. Appl.25, 33-49 (1991). Zbl0809.47044
- [42] P. Sarnak: Arithmetic quantum chaos, Israel Math. Conf. Proceedings, 8, 183-236 (1995). Zbl0831.58045
- [43] K. Takasaki, T. Takebe : Integrable hierarchies and dispersionless limit, Reviews in Math. Phys.7, 743-808 (1995) . Zbl0838.35117
- [44] C.A. Tracy, H. Widom: Level-spacings distribution and the Airy kernel, Comm. Math. Phys.159, 151-174 (1984). Zbl0789.35152
- [45] K. Ueno, K. Takasaki : Toda Lattice Hierarchy, Adv. Studies in Pure Math.4, 1-95 (1984). Zbl0577.58020
- [46] J. van de Leur: The W1+∞(gls)-symmetries of the s-component KP hierarchy, J. of Math. Phys.37, 2315-2337 (1996). Zbl0864.35096
- [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).
- [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). Zbl0044.44203
- [51] Witten, Ed. : Two-dimensional gravity and intersection theory of moduli space, Harvard University lecture, May 1990, Journal of diff. geometry, 1991. Zbl0757.53049
- [52] V. Zakharov, L.D. Faddeev: The KdV equation is a completely integrable Hamiltonian system, Funct. Anal. Appl.5, 18-27 (1971). Zbl0257.35074
- [53] A.B. Zamolodchikov: Infinite additional symmetries in two-dimensional conformal quantum field theory, Theoret. Math. Phys.65, 1205 (1985).
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