Bispectral Symmetry, the Weyl Algebra and Differential Operators on Curves
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 153-159.

Voir la notice de l'article provenant de la source Math-Net.Ru

A theorem of Cannings and Holland can be interpreted as identifying the space of rational solutions to the KP hierarchy with the space of isomorphism classes of right ideals in the Weyl algebra. The article sketches some applications of this connection.
@article{TM_1999_225_a8,
     author = {G. Wilson},
     title = {Bispectral {Symmetry,} the {Weyl} {Algebra} and {Differential} {Operators} on {Curves}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {153--159},
     publisher = {mathdoc},
     volume = {225},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_225_a8/}
}
TY  - JOUR
AU  - G. Wilson
TI  - Bispectral Symmetry, the Weyl Algebra and Differential Operators on Curves
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 1999
SP  - 153
EP  - 159
VL  - 225
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_1999_225_a8/
LA  - en
ID  - TM_1999_225_a8
ER  - 
%0 Journal Article
%A G. Wilson
%T Bispectral Symmetry, the Weyl Algebra and Differential Operators on Curves
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 1999
%P 153-159
%V 225
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_1999_225_a8/
%G en
%F TM_1999_225_a8
G. Wilson. Bispectral Symmetry, the Weyl Algebra and Differential Operators on Curves. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 153-159. http://geodesic.mathdoc.fr/item/TM_1999_225_a8/

[1] Artamkin I. V., “Deistvie biregulyarnykh avtomorfizmov affinnoi ploskosti na pary matrits”, Izv. AN SSSR. Ser. mat., 52:5 (1988), 1109–1115 | MR

[2] Bakalov B., Horozov E., Yakimov M., “Automorphisms of the Weyl algebra and bispectral operators”, The bispectral problem (Montreal, PQ, 1997), CRM Proc. Lecture Notes, 14, Amer. Math. Soc., Providence, R.I., 1998, 3–10 | MR | Zbl

[3] Berest Yu., Wilson G., “Classification of rings of differential operators on affine curves”, Intern. Math. Res. Notic., 1999, no. 2, 105–109 | DOI | MR | Zbl

[4] Cannings R. C., Holland M. P., “Right ideals in rings of differential operators”, J. Algebra, 167 (1994), 116–141 | DOI | MR | Zbl

[5] Dixmier J., “Sur les algèbres de Weyl”, Bull. Soc. math. France, 96 (1968), 209–242 | MR | Zbl

[6] Duistermaat J. J., Grünbaum F. A., “Differential equations in the spectral parameter”, Communs Math. Phys., 103 (1986), 177–240 | DOI | MR | Zbl

[7] Kouakou K. M., Isomorphismes entre algèbres d'opérateurs différentiels sur les courbes algébriques affines, Thèse. Univ. Claude Bernard Lyon-1, 1994

[8] Krichever I. M., “On rational solutions of the Kadomtsev–Petviashvili equation and integrable systems of $N$ particles on the line”, Funct. Anal. Appl., 12:1 (1978), 59–61 | MR | Zbl

[9] Kazhdan D., Kostant B., Sternberg S., “Hamiltonian group actions and dynamical systems of Calogero type”, Commun. Pure and Appl. Math., 31 (1978), 481–507 | DOI | MR | Zbl

[10] Le Bruyn L., “Moduli spaces for right ideals of the Weyl algebra”, J. Algebra, 172 (1995), 32–48 | DOI | MR | Zbl

[11] Letzter G., Makar-Limanov L., “Rings of differential operators over rational affine curves”, Bull. Soc. math. France, 118 (1990), 193–209 | MR | Zbl

[12] Makar-Limanov L., “On automorphisms of Weyl algebra”, Bull. Soc. math. France, 112 (1984), 359–363 | MR | Zbl

[13] Makar-Limanov L., “Rings of differential operators on algebraic curves”, Bull. London Math. Soc., 21 (1989), 538–540 | DOI | MR | Zbl

[14] Nakajima H., Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, American Mathematical Society, Providence, R.I., 1999 | MR | Zbl

[15] Perkins P., “Commutative subalgebras of the ring of differential operators on a curve”, Pacif. J. Math., 139:2 (1989), 279–302 | MR | Zbl

[16] Sato M., “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds”, Nonlinear partial differential equations in applied science (Tokyo, 1982), North-Holland Math. Stud., 81, North-Holland, Amsterdam, 1983, 259–271 | MR | Zbl

[17] Shafarevich I. R., “O nekotorykh beskonechnomernykh gruppakh, II”, Izv. AN SSSR. Ser. mat., 45:1 (1981), 214–226 | MR | Zbl

[18] Stafford J. T., “Endomorphisms of right ideals in the Weyl algebra”, Trans. Amer. Math. Soc., 299 (1987), 624–639 | DOI | MR

[19] Wilson G., “Bispectral commutative ordinary differential operators”, J. reine und angew. Math., 442 (1993), 177–204 | MR | Zbl

[20] Wilson G., “Collisions of Calogero–Moser particles and an adelic Grassmannian”, Invent. math., 133 (1998), 1–41 ; I. G. Macdonald, “Appendix to Wilson G.” | DOI | MR | Zbl

[21] Wilson G., “Bispectral algebras of ordinary differential operators”, The bispectral problem (Montreal, PQ, 1997), CRM Proc. Lecture Notes, 14, Amer. Math. Soc., Providence, R.I., 1998, 131–137 | MR | Zbl

[22] Wilson G., “The complex Calogero–Moser and KP systems”, Calogero-Moser-Sutherland models (Montréal, QC, 1997), CRM Ser. Math. Phys., Springer, New York, 2000, 539–548 | MR