Calogero-Moser spaces and an adelic W-algebra
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, p. 2069-2090

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.

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.

DOI : https://doi.org/10.5802/aif.2152
Classification:  37K30,  37K35
Keywords: Fock spaces, bispectral operators, Sato's theory for KP hierarchy
@article{AIF_2005__55_6_2069_0,
     author = {Horozov, Emil},
     title = {Calogero-Moser spaces and an adelic $W$-algebra},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     pages = {2069-2090},
     doi = {10.5802/aif.2152},
     zbl = {02230068},
     mrnumber = {2187946},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_6_2069_0}
}
Horozov, Emil. Calogero-Moser spaces and an adelic $W$-algebra. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 2069-2090. doi : 10.5802/aif.2152. http://www.numdam.org/item/AIF_2005__55_6_2069_0/

[1] M. Adler; J. Moser On a class of polynomials connected with the Korteweg-de Vries equation, Commun. Math. Phys., Tome 61 (1978), pp. 1-30 | Article | MR 501106 | Zbl 0428.35067

[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., Tome 30 (1977), pp. 95-148 | Article | MR 649926 | Zbl 0338.35024

[3] M. Adler; T. Shiota; P. Van Moerbeke A Lax representation for the vertex operator and the central extension, Commun. Math. Phys., Tome 171 (1995), pp. 547-588 | Article | MR 1346172 | Zbl 0839.35116

[4] B.N. Bakalov; L.S. Georgiev; I.T. Todorov; A. Ganchev Et Al. A QFT approach to W 1+ , New Trends in Quantum Field Theory, Proc. of the 1995 Razlog (Bulgaria) Workshop, Heron Press, Sofia (1996), pp. 147-158

[5] B. Bakalov; E. Horozov; M. Yakimov Tau-functions as highest weight vectors for W 1+ algebra, J. Phys. A. Math. Gen., Tome 29 (1996), pp. 5565-5573 | Article | MR 1419041 | Zbl 0906.17018

[6] B. Bakalov; E. Horozov; M. Yakimov Bäcklund-Darboux transformations in Sato's Grassmannian, Serdica Math. J., Tome 4 (1996) | MR 1483606 | Zbl 0934.37016

[7] B. Bakalov; E. Horozov; M. Yakimov Bispectral algebras of commuting ordinary differential operators, Comm. Mat. Phys., Tome 190 (1997), pp. 331-373 | Article | MR 1489575 | Zbl 0912.34065

[8] B. Bakalov; E. Horozov; M. Yakimov Highest weight modules over W 1+ , and the bispectral problem, Duke Math. J., Tome 93 (1998), pp. 41-72 | Article | MR 1620079 | Zbl 0983.17015

[9] Yu. Berest; G. Wilson Automorphisms and ideals of the Weyl algebra, Math. Ann., Tome 318 (2000) no. 1, pp. 127-147 | Article | MR 1785579 | Zbl 0983.16021

[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 1055.16030

[11] R.C. Cannings; M.P. Holland Right ideals in rings of differential operators, J. Algebra, Tome 167 (1994), pp. 116-141 | Article | MR 1282820 | Zbl 0824.16022

[12] F. Calogero Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., Tome 12 (1971), pp. 419-436 | Article | MR 280103 | Zbl 1002.70558

[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), Singapore: World Scientific (1983), pp. 39-111 | Zbl 0571.35098

[14] L. Dickey Soliton equations and integrable systems, Singapore: World Scientific (1991) | MR 1147643 | Zbl 0753.35075

[15] J.J. Duistermaat; F.A. Grünbaum Differential equations in the spectral parameter, Commun. Math. Phys., Tome 103 (1986), pp. 177-240 | Article | MR 826863 | Zbl 0625.34007

[16] B. Fuchsteiner Master-symmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theor. Phys., Tome 70 (1983) no. 6, pp. 1508-1522 | Article | MR 734645 | Zbl 01662637

[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, AMS (Proc. Symp. Appl. Math.) Tome 27 (1982), pp. 43-61 | Zbl 0536.65094

[19] F.A. Grünbaum; L. Haine; E. Horozov Some functions that generalize the Krall-Laguerre polynomials, J. Comp. Appl. Math., Tome 106 (1999) no. 2, pp. 271-297 | Article | MR 1696411 | Zbl 0926.33007

[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 1051.39020

[21] L. Haine; P. Iliev Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Notices, Tome 6 (2000), pp. 281-323 | MR 1749073 | Zbl 0984.37078

[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 1916779 | Zbl 1037.47030

[23] E. Horozov The Weyl algebra, bispectral operators and dynamics of poles in integrable systems, Reg. \& Chaotic Dynamics, Tome 7 (2002) no. 4, pp. 399-424 | Article | MR 1957273 | Zbl 1034.34100

[24] Pl. Iliev Algèbres commutatives d'opérateurs aux q-differences et systèmes de Calogero-Moser, C. R. Sci. Paris, Série I, Tome 329 (1999), pp. 877-882 | MR 1728001 | Zbl 0947.39010

[25] D. Kazhdan; B. Kostant; S. Sternberg Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math., Tome 31 (1978), pp. 481-507 | Article | MR 478225 | Zbl 0368.58008

[26] I.M Krichever On Rational solutions of Kadomtsev-Petviashvily equation and integrable systems of N particles on the line, Funct. Anal. Appl., Tome 12 (1978) no. 1, pp. 76-78 | Zbl 0408.70010

[27] V.G. Kac; D.H. Peterson Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA, Tome 78 (1981), pp. 3308-3312 | Article | MR 619827 | Zbl 0469.22016

[28] V.G. Kac; A. Radul Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys., Tome 9308153 (1993) no. 157, pp. 429-457 | MR 1243706 | Zbl 0826.17027

[29] V.G. Kac; A. Raina Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Singapore: World Scientific, Adv. Ser. Math. Phys., Tome 2 (1987) | Zbl 0668.17012

[30] A. Kasman Bispectral KP solutions and linearization of Calogero-Moser particle systems, Commun. Math. Phys., Tome 172 (1995), pp. 427-448 | Article | MR 1350415 | Zbl 0842.58047

[31] J. Moser Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., Tome 16 (1975), pp. 197-220 | Article | MR 375869 | Zbl 0303.34019

[32] A.Yu. Orlov; Baryakhtar Vertex operators, ¯-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, Singapore: World Scientific (1988) | Zbl 0691.35075

[33] A.Yu. Orlov; E.I. Schulman Additional symmetries for integrable and conformal algebra representation, Lett. Math. Phys., Tome 12 (1989), pp. 171-179 | Zbl 0618.35107

[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, CRM Proceedings and Lecture Notes, AMS (1998), pp. Providence | Zbl 0985.37081

[35] M. Sato Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku, Tome 439 (1981), pp. 30-40 | Zbl 0507.58029

[36] G. Segal; G. Wilson Loop Groups and equations of KdV type, Publ. Math. IHES, Tome 61 (1985), pp. 5-65 | Numdam | MR 783348 | Zbl 0592.35112

[37] P. Van Moerbeke; O. Babelon Et Al. Integrable foundations of string theory, Singapore: World Scientific (CIMPA-Summer school at Sophia-Antipolis (1991), in: Lectures on integrable systems) (1994), pp. 163-267 | Zbl 0850.81049

[38] G. Wilson Bispectral commutative ordinary differential operators, J. Reine Angew. Math., Tome 442 (1993), pp. 177-204 | MR 1234841 | Zbl 0781.34051

[39] G. Wilson Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald), Invent. Math., Tome 133 (1998), pp. 1-41 | MR 1626461 | Zbl 0906.35089