Calogero-Moser spaces and an adelic -algebra
Emil Horozov[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 (2005)
- Volume: 55, Issue: 6, page 2069-2090
- ISSN: 0373-0956
Access Full Article
topAbstract
topHow to cite
topHorozov, Emil. "Calogero-Moser spaces and an adelic $W$-algebra." Annales de l’institut Fourier 55.6 (2005): 2069-2090. <http://eudml.org/doc/116244>.
@article{Horozov2005,
abstract = {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.},
affiliation = {Bulgarian Academy of Science, institute of mathematics and informatics, acad. G. Bonchev Str., Block 8, 1113 Sofia (Bulgarie)},
author = {Horozov, Emil},
journal = {Annales de l’institut Fourier},
keywords = {Fock spaces; bispectral operators; Sato's theory for KP hierarchy},
language = {eng},
number = {6},
pages = {2069-2090},
publisher = {Association des Annales de l'Institut Fourier},
title = {Calogero-Moser spaces and an adelic $W$-algebra},
url = {http://eudml.org/doc/116244},
volume = {55},
year = {2005},
}
TY - JOUR
AU - Horozov, Emil
TI - Calogero-Moser spaces and an adelic $W$-algebra
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 6
SP - 2069
EP - 2090
AB - 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.
LA - eng
KW - Fock spaces; bispectral operators; Sato's theory for KP hierarchy
UR - http://eudml.org/doc/116244
ER -
References
top- M. Adler, J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Commun. Math. Phys. 61 (1978), 1-30 Zbl0428.35067MR501106
- 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. 30 (1977), 95-148 Zbl0338.35024MR649926
- M. Adler, T. Shiota, P. van Moerbeke, A Lax representation for the vertex operator and the central extension, Commun. Math. Phys. 171 (1995), 547-588 Zbl0839.35116MR1346172
- B.N. Bakalov, L.S. Georgiev, I.T. Todorov, A QFT approach to , New Trends in Quantum Field Theory, Proc. of the 1995 Razlog (Bulgaria) Workshop (1996), 147-158, Heron Press, Sofia
- B. Bakalov, E. Horozov, M. Yakimov, Tau-functions as highest weight vectors for algebra, J. Phys. A. Math. Gen. 29 (1996), 5565-5573 Zbl0906.17018MR1419041
- B. Bakalov, E. Horozov, M. Yakimov, Bäcklund-Darboux transformations in Sato's Grassmannian, Serdica Math. J. 4 (1996) Zbl0934.37016MR1483606
- B. Bakalov, E. Horozov, M. Yakimov, Bispectral algebras of commuting ordinary differential operators, Comm. Mat. Phys. 190 (1997), 331-373 Zbl0912.34065MR1489575
- B. Bakalov, E. Horozov, M. Yakimov, Highest weight modules over , and the bispectral problem, Duke Math. J. 93 (1998), 41-72 Zbl0983.17015MR1620079
- Yu. Berest, G. Wilson, Automorphisms and ideals of the Weyl algebra, Math. Ann. 318 (2000), 127-147 Zbl0983.16021MR1785579
- Yu. Berest, G. Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry, (2001) Zbl1055.16030
- R.C. Cannings, M.P. Holland, Right ideals in rings of differential operators, J. Algebra 167 (1994), 116-141 Zbl0824.16022MR1282820
- F. Calogero, Solution of the one-dimensional -body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436 Zbl1002.70558MR280103
- 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) (1983), 39-111, Singapore: World Scientific Zbl0571.35098
- L. Dickey, Soliton equations and integrable systems, Singapore: World Scientific (1991) Zbl0753.35075MR1147643
- J.J. Duistermaat, F.A. Grünbaum, Differential equations in the spectral parameter, Commun. Math. Phys. 103 (1986), 177-240 Zbl0625.34007MR826863
- B. Fuchsteiner, Master-symmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theor. Phys. 70 (1983), 1508-1522 Zbl1098.37536MR734645
- P.G. Grinevich, A.Yu. Orlov, E.I. Schulman, On the symmetries of the integrable system, Modern development of the Soliton theory (1992)
- P.A. Grünbaum, The limited angle reconstruction problem in computer tomography, 27 (1982), 43-61, AMS Zbl0536.65094
- F.A. Grünbaum, L. Haine, E. Horozov, Some functions that generalize the Krall-Laguerre polynomials, J. Comp. Appl. Math. 106 (1999), 271-297 Zbl0926.33007MR1696411
- F.A. Grünbaum, M. Yakimov, Discrete bispectral Darboux transformations from Jacobi operators, (2000) Zbl1051.39020
- L. Haine, P. Iliev, Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Notices 6 (2000), 281-323 Zbl0984.37078MR1749073
- 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) Zbl1037.47030MR1916779
- E. Horozov, The Weyl algebra, bispectral operators and dynamics of poles in integrable systems, Reg. & Chaotic Dynamics 7 (2002), 399-424 Zbl1034.34100MR1957273
- Pl. Iliev, Algèbres commutatives d’opérateurs aux -differences et systèmes de Calogero-Moser, C. R. Sci. Paris, Série I 329 (1999), 877-882 Zbl0947.39010MR1728001
- D. Kazhdan, B. Kostant, S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481-507 Zbl0368.58008MR478225
- I.M Krichever, On Rational solutions of Kadomtsev-Petviashvily equation and integrable systems of particles on the line, Funct. Anal. Appl. 12 (1978), 76-78 Zbl0408.70010
- V.G. Kac, D.H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA 78 (1981), 3308-3312 Zbl0469.22016MR619827
- V.G. Kac, A. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys. 9308153 (1993), 429-457 Zbl0826.17027MR1243706
- V.G. Kac, A. Raina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, 2 (1987), Singapore: World Scientific Zbl0668.17012
- A. Kasman, Bispectral KP solutions and linearization of Calogero-Moser particle systems, Commun. Math. Phys. 172 (1995), 427-448 Zbl0842.58047MR1350415
- J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220 Zbl0303.34019MR375869
- A.Yu. Orlov, Vertex operators, -problem, symmetries, variational identities and Hamiltonian formalism for integrable systems, Proc. Kiev Intern. Workshop, Plasma theory and non-linear and turbulent processes in Physics (1988), Singapore: World Scientific Zbl0691.35075
- A.Yu. Orlov, E.I. Schulman, Additional symmetries for integrable and conformal algebra representation, Lett. Math. Phys. 12 (1989), 171-179 Zbl0618.35107
- M. Rothstein, Explicit formulas for the Airy and Bessel involutions in terms of Calogero-Moser pairs, The Bispectral Problem (1998), CRM Proceedings and Lecture Notes, AMS Zbl0985.37081
- M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku 439 (1981), 30-40 Zbl0507.58029
- G. Segal, G. Wilson, Loop Groups and equations of KdV type, Publ. Math. IHES 61 (1985), 5-65 Zbl0592.35112MR783348
- P. van Moerbeke, Integrable foundations of string theory, (1994), 163-267, Singapore: World Scientific Zbl0850.81049
- G. Wilson, Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177-204 Zbl0781.34051MR1234841
- G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald), Invent. Math. 133 (1998), 1-41 Zbl0906.35089MR1626461
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.