Calogero-Moser spaces and an adelic $W$-algebra

Horozov, Emil

doi:10.5802/aif.2152

Calogero-Moser spaces and an adelic

W

-algebra
[Espaces de Calogero-Moser et une

W

-algèbre adélique]

Horozov, Emil ¹

¹ Bulgarian Academy of Science, institute of mathematics and informatics, acad. G. Bonchev Str., Block 8, 1113 Sofia (Bulgarie)

Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2069-2090.

Résumé
Abstract

Nous construisons une algèbre nommée adélique $W$ -algèbre puis, nous construisons une représentation bosonique naturelle. Nous montrons ensuite que les points des espaces de Calogero-Moser sont en correspondance biunivoque avec les fonctions tau en cette représentation.

We introduce a Lie algebra, which we call adelic $W$ -algebra. Then we construct a natural bosonic representation and show that the points of the Calogero-Moser spaces are in 1:1 correspondence with the tau-functions in this representation.

MR Zbl

DOI : 10.5802/aif.2152

Classification : 37K30, 37K35
Keywords: Fock spaces, bispectral operators, Sato's theory for KP hierarchy
Mot clés : espaces de Fock, opérateurs bispectraux, théorie de Sato de KP-hiérarchie

Affiliations des auteurs :

Horozov, Emil ¹

¹ Bulgarian Academy of Science, institute of mathematics and informatics, acad. G. Bonchev Str., Block 8, 1113 Sofia (Bulgarie)

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Horozov, Emil. Calogero-Moser spaces and an adelic $W$-algebra. Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2069-2090. doi : 10.5802/aif.2152. https://www.numdam.org/articles/10.5802/aif.2152/

Bibliographie
Cité par

[1] M. Adler; J. Moser On a class of polynomials connected with the Korteweg-de Vries equation, Commun. Math. Phys., Volume 61 (1978), pp. 1-30 | DOI | MR | Zbl

[2] H. Airault; H.P. McKean; J. Moser Rational and elliptic solutions to the Korteweg-de Vries equation and related many-body problem, Comm. Pure Appl. Math., Volume 30 (1977), pp. 95-148 | DOI | MR | Zbl

[3] M. Adler; T. Shiota; P. van Moerbeke A Lax representation for the vertex operator and the central extension, Commun. Math. Phys., Volume 171 (1995), pp. 547-588 | DOI | MR | Zbl

[4] B.N. Bakalov; L.S. Georgiev; I.T. Todorov; A. Ganchev et al. A QFT approach to $W_{1 + \infty}$ , New Trends in Quantum Field Theory, Proc. of the 1995 Razlog (Bulgaria) Workshop (1996), pp. 147-158

[5] B. Bakalov; E. Horozov; M. Yakimov Tau-functions as highest weight vectors for $W_{1 + \infty}$ algebra, J. Phys. A. Math. Gen., Volume 29 (1996), pp. 5565-5573 | DOI | MR | Zbl

[6] B. Bakalov; E. Horozov; M. Yakimov Bäcklund-Darboux transformations in Sato's Grassmannian, Serdica Math. J., Volume 4 (1996) | MR | Zbl

[7] B. Bakalov; E. Horozov; M. Yakimov Bispectral algebras of commuting ordinary differential operators, Comm. Mat. Phys., Volume 190 (1997), pp. 331-373 | DOI | MR | Zbl

[8] B. Bakalov; E. Horozov; M. Yakimov Highest weight modules over $W_{1 + \infty}$ , and the bispectral problem, Duke Math. J., Volume 93 (1998), pp. 41-72 | DOI | MR | Zbl

[9] Yu. Berest; G. Wilson Automorphisms and ideals of the Weyl algebra, Math. Ann., Volume 318 (2000) no. 1, pp. 127-147 | DOI | MR | Zbl

[10] Yu. Berest; G. Wilson Ideal classes of the Weyl algebra and noncommutative projective geometry (2001) (arXiv.math.AG/0104248, http://arxiv.org/abs/math.AG/0104248) | Zbl

[11] R.C. Cannings; M.P. Holland Right ideals in rings of differential operators, J. Algebra, Volume 167 (1994), pp. 116-141 | DOI | MR | Zbl

[12] F. Calogero Solution of the one-dimensional $n$ -body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., Volume 12 (1971), pp. 419-436 | DOI | MR | Zbl

[13] E. Date; M. Jimbo; M. Kashiwara; T. Miwa; M. Jimbo, T. Miwa Transformation groups for soliton equations, Proc. RIMS Symp., Nonlinear integrable systems - Classical and Quantum theory, (Kyoto 1981) (1983), pp. 39-111 | Zbl

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

[15] J.J. Duistermaat; F.A. Grünbaum Differential equations in the spectral parameter, Commun. Math. Phys., Volume 103 (1986), pp. 177-240 | DOI | MR | Zbl

[16] B. Fuchsteiner Master-symmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theor. Phys., Volume 70 (1983) no. 6, pp. 1508-1522 | DOI | MR | Zbl

[17] P.G. Grinevich; A.Yu. Orlov; E.I. Schulman; A. Fokas, V.E. Zakharov On the symmetries of the integrable system, Modern development of the Soliton theory (1992)

[18] P.A. Grünbaum; L. Shepp The limited angle reconstruction problem in computer tomography (Proc. Symp. Appl. Math.), Volume 27 (1982), pp. 43-61 | Zbl

[19] F.A. Grünbaum; L. Haine; E. Horozov Some functions that generalize the Krall-Laguerre polynomials, J. Comp. Appl. Math., Volume 106 (1999) no. 2, pp. 271-297 | DOI | MR | Zbl

[20] F.A. Grünbaum; M. Yakimov Discrete bispectral Darboux transformations from Jacobi operators (2000) (arXiv.mat.CA/0012191, http://arxiv.org/abs/math.CA/0012191) | Zbl

[21] L. Haine; P. Iliev Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Notices, Volume 6 (2000), pp. 281-323 | MR | Zbl

[22] E. Horozov Dual algebras of differential operators, in: Kowalevski property (Montréal), CRM Proc. Lecture Notes, Surveys from Kowalevski Workshop on Mathematical methods of Regular Dynamics, Leeds, April 2000 (2002) no. Amer. Math. Soc. Providence | MR | Zbl

[23] E. Horozov The Weyl algebra, bispectral operators and dynamics of poles in integrable systems, Reg. \& Chaotic Dynamics, Volume 7 (2002) no. 4, pp. 399-424 | DOI | MR | Zbl

[24] Pl. Iliev Algèbres commutatives d'opérateurs aux $q$ -differences et systèmes de Calogero-Moser, C. R. Sci. Paris, Série I, Volume 329 (1999), pp. 877-882 | MR | Zbl

[25] D. Kazhdan; B. Kostant; S. Sternberg Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., Volume 31 (1978), pp. 481-507 | DOI | MR | Zbl

[26] I.M Krichever On Rational solutions of Kadomtsev-Petviashvily equation and integrable systems of $N$ particles on the line, Funct. Anal. Appl., Volume 12 (1978) no. 1, pp. 76-78 | Zbl

[27] V.G. Kac; D.H. Peterson Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA, Volume 78 (1981), pp. 3308-3312 | DOI | MR | Zbl

[28] V.G. Kac; A. Radul Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys., Volume 9308153 (1993) no. 157, pp. 429-457 | MR | Zbl

[29] V.G. Kac; A. Raina Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Adv. Ser. Math. Phys., 2, Singapore: World Scientific, 1987 | Zbl

[30] A. Kasman Bispectral KP solutions and linearization of Calogero-Moser particle systems, Commun. Math. Phys., Volume 172 (1995), pp. 427-448 | DOI | MR | Zbl

[31] J. Moser Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., Volume 16 (1975), pp. 197-220 | DOI | MR | Zbl

[32] A.Yu. Orlov; Baryakhtar Vertex operators, $\bar{\partial}$ -problem, symmetries, variational identities and Hamiltonian formalism for $2 + 1$ integrable systems, Proc. Kiev Intern. Workshop, Plasma theory and non-linear and turbulent processes in Physics (1988) | Zbl

[33] A.Yu. Orlov; E.I. Schulman Additional symmetries for integrable and conformal algebra representation, Lett. Math. Phys., Volume 12 (1989), pp. 171-179 | Zbl

[34] M. Rothstein; J. Harnad and A. Kasman Explicit formulas for the Airy and Bessel involutions in terms of Calogero-Moser pairs, The Bispectral Problem (1998), pp. Providence | Zbl

[35] M. Sato Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku, Volume 439 (1981), pp. 30-40 | Zbl

[36] G. Segal; G. Wilson Loop Groups and equations of KdV type, Publ. Math. IHES, Volume 61 (1985), pp. 5-65 | Numdam | MR | Zbl

[37] P. van Moerbeke; O. Babelon et al. Integrable foundations of string theory (CIMPA-Summer school at Sophia-Antipolis (1991), in: Lectures on integrable systems) (1994), pp. 163-267 | Zbl

[38] G. Wilson Bispectral commutative ordinary differential operators, J. Reine Angew. Math., Volume 442 (1993), pp. 177-204 | MR | Zbl

[39] G. Wilson Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald), Invent. Math., Volume 133 (1998), pp. 1-41 | MR | Zbl

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