×
Adams, M. R.; Bergvelt, M. J.

The Krichever map, vector bundles over algebraic curves, and Heisenberg algebras. (English) Zbl 0787.35085

Commun. Math. Phys. 154, No. 2, 265-305 (1993).
The authors study an interconnection between two approaches in the theory of integrable nonlinear evolution equations. In the algebraic geometry approach one starts with an algebraic curve and then builds solutions of the integrable evolution equations in terms of theta-functions identifying these solutions later with points of infinite dimensional Grassmannian or, which is equivalent, with orbits of appropriate infinite dimensional groups. In the group theory approach one begins with a Kac- Moody algebra, a Heisenberg subalgebra and an integrated irreducible representation in terms of vertex operators and then constructs an orbit of the associated group through the highest weight vector connecting later the defining equation of the orbit with the integrable nonlinear evolution equation.
The authors investigate how in the framework of the latter approach it is possible for every point of the Grassmannian and every choice of a Heisenberg algebra to build an algebraic curve in terms of a cotangent bundle of the Grassmannian. In particular they study the significance of the choice of the Heisenberg algebra for the construction of the algebraic curve.
Reviewer: E.D.Belokolos (Kiev)

Cited in 1 Review
Cited in 6 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
14H42 Theta functions and curves; Schottky problem
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Keywords:

integrable nonlinear evolution equations; algebraic geometry approach; theta-functions; Grassmannian; group theory approach; Heisenberg subalgebra; irreducible representation; cotangent bundle
Cite Review PDF
Full Text: DOI

References:

[1] [AB] Adams, M.R., Bergvelt, M.J.: Heisenberg algebras, Grassmannians and isospectral curves, The Geometry of Hamiltonian Systems. MSRI Publication 22, Ratiu, T. (ed.) Berlin, Heidelberg, New York: Springer 1991, pp. 1–8 · Zbl 0752.17022
[2] [AHH] Adams, M.R., Harnad, J., Hurtubise, J.: Coadjoint orbits, spectral curves and Darboux coordinates. The Geometry of Hamiltonian Systems, MSRI Publication 22, Ratiu, T. (ed.) Berlin, Heidelberg, New York: Springer 1991, pp. 9–22 · Zbl 0739.58014
[3] [Ad] Adler, M.: On a Trace Functional for Formal Pseudo-Differential Operators and the Symplectic Structure of the Korteweg-de Vries Equation. Invent. Math.50, 219–248 (1979) · Zbl 0393.35058 · doi:10.1007/BF01410079
[4] [ADKP] Arbarello, E., De Concini, C., Kac, V., Procesi, C.: Moduli spaces of curves and representation theory. Commun. Math. Phys.117, 1–36 (1988) · Zbl 0647.17010 · doi:10.1007/BF01228409
[5] [AtB] Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Phil. Trans. R. Soc. Lond. A308, 523–615 (1982) · Zbl 0509.14014
[6] [B] Beauville, A.: Jacobiennes des courbes spectrales et systemes Hamiltoniens completement integrables. Preprint (1989)
[7] [BNR] Beauville, A., Narasimhan, M.S., Ramanan, S.: Spectral curves and the generalized theta divisor. J. fur die reine und angewandte. Math.398, 169–179 (1989) · Zbl 0666.14015
[8] [BtK] Bergvelt, M.J., ten Kroode, A.P.E.: Partitions, vertex operator constructions and multicomponent KP equations. Preprint (1991)
[9] [Bo] Bourbaki, N.: Commutative algebra. Berlin, Heidelberg, New York: Springer 1989 · Zbl 0666.13001
[10] [Br] Brylinski, J.L.: Loop groups and non-commutative theta functions. Preprint (1989)
[11] [CEH] Carey, A.L., Eastwood, M.G., Hannabus, K.C.: Riemann surfaces, Clifford algebras and infinite dimensional groups. Commun. Math. Phys.130, 217–236 (1990) · Zbl 0721.30034 · doi:10.1007/BF02473351
[12] [Che] Cherednik, I.V.: Definitions of functions for affine Lie algebras. Funct. Anal. Appl.17, 243 (1983); On the group-theoretical interpretation of Baker functions and {\(\tau\)}-functions. Russ. Math. Surveys38-6, 113–114 · Zbl 0528.17004 · doi:10.1007/BF01078119
[13] [DaJKM] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. Proceedings of RIMS symposium, Jimbo, M., Miwa, T. (eds.) Singapore: World Scientific 1983, pp. 39–120 · Zbl 0571.35098
[14] [DaJM] Date, E., Jimbo, M., Miwa, T.: Landau-Lifshitz equation. J. Phys. Soc. A1983, 388–393 (1983) · Zbl 0571.35105 · doi:10.1143/JPSJ.52.388
[15] [Di] Dickey, L.A.: Another example of a {\(\tau\)}-function. Proceedings of the Workshop on Hamiltonian systems. Montreal (1989) Harnad, J., Marsden, J.E. (eds.) Les Publications CRM, Montreal, 1990, pp. 39–44; On the {\(\tau\)}-functions of matrix hierarchies of integrable equations; On Segal-Wilson’s definition of the {\(\tau\)}-function and hierarchies AKNS-D and mcKP
[16] [DS] Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of KdV-type. J. Sov. Math.30, 1975–2036 (1985) · Zbl 0578.58040 · doi:10.1007/BF02105860
[17] [Du] Dubrovin, B.A.: Theta Functions and Nonlinear Equations. Russ. Math. Surveys35, 11–92 (1981) · Zbl 0549.58038 · doi:10.1070/RM1981v036n02ABEH002596
[18] [FNR] Flaschka, H., Newell, A.C., Ratiu, T.: Kac Moody algebras and solution equations, II, Lax equation associated withA 1 (1) . Physica9D, 300 (1983) · Zbl 0643.35098
[19] [GD] Gelfand, I.M., Dickey, L.A.: Fractional powers of operators and Hamiltonian systems. Funct. Anal. Appl.10, 259–273 (1976); The resolvent and Hamiltonian systems. Funct. Anal. Appl.11, 93–105 (1977) · Zbl 0356.35072 · doi:10.1007/BF01076025
[20] [GH] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley 1978 · Zbl 0408.14001
[21] [GS1] Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge: Cambridge University Press 1984 · Zbl 0576.58012
[22] [GS2] Guillemin, V., Sternberg, S.: On the method of Symes for integrating systems of the Toda type. Lett. Math. Phys.7, 113–115 (1983) · Zbl 0521.58033 · doi:10.1007/BF00419928
[23] [H] Harder, G.: Eine Bemerkung zu einer Arbeit von P.E. Newstaed. J. fur Mathematik242, 16–25 (1970) · Zbl 0219.14016
[24] [Ha] Hartshorne, R.: Algebraic geometry. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0367.14001
[25] [Hi] Hitchin, N.: Stable bundles and integrable systems. Duke Math. J.54, 91–114 (1987) · Zbl 0627.14024 · doi:10.1215/S0012-7094-87-05408-1
[26] [Hi2] Hitchin, N.: Harmonic maps from the 2-torus to the 3-sphere. J. Diff. Geom.31, 627–710 (1990) · Zbl 0725.58010
[27] [Ka] Kac, V.G.: Infinite dimensional Lie algebras. Second edition, Boston: Cambridge University Press 1985 · Zbl 0574.17010
[28] [KaP] Kac, V.G., Peterson, D.H.: 112 constructions of the basic representation of the loopgroups ofE 8, Proceedings of the symposium on anomalies, geometry and topology, Singapore: World Scientific 1985, pp. 276–298
[29] [KaW] Kac, V.G., Wakimoto, M.: Exceptional hierarchies of soliton equations. Theta functions, Bowdoin 1987, vol.49, Ehrenpreis, L., Gunning, R.C. (eds.) Proceedings of symposia in pure mathematics. Am. Math. Soc., 1989, pp. 191–237
[30] [KaSU] Katsura, T., Shimizu, Y., Ueno, K.: New Bosonization and Conformal Field Theory over Z. Commun. Math. Phys.121, 603–627 (1989) · Zbl 0672.14016 · doi:10.1007/BF01218158
[31] [KNTY] Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y.: Geometric realization of conformal field theory on Riemann surfaces Commun. Math. Phys.116, 247–308 (1988) · Zbl 0648.35080 · doi:10.1007/BF01225258
[32] [KdV] Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and a new type of long stationary waves. Phil. Mag.39, 422–443 (1895) · JFM 26.0881.02
[33] [KS] Kossmann-Schwarzbach, Y.: Quasi-bigebres de Lie et groupes de Lie quasi-Poisson. C.R. Acad. Sci. Paris312, Serie I, 391–394 (1991) · Zbl 0712.22012
[34] [Ko] Kostant, B.: The solution of the generalized Toda lattice and representation theory. Adv. Math.34, 195–338 (1979) · Zbl 0433.22008 · doi:10.1016/0001-8708(79)90057-4
[35] [Kr] Krichever, I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Russ. Math. Surv.32, 185–213 (1977) · Zbl 0386.35002 · doi:10.1070/RM1977v032n06ABEH003862
[36] [KrN] Krichever, I.M., Novikov, S.P.: Holomorphic Bundles over Algebraic Curves and Nonlinear Equations. Russ. Math. Surv.35 (1980) · Zbl 0548.35100
[37] [tKr] ten Kroode, A.P.E.: Affine Lie algebras and integrable systems. Thesis, University of Amsterdam 1988
[38] [Ku] Kunz, E.: Introduction to commutative algebra and algebraic geometry. Boston: Birkhäuser 1985 · Zbl 0563.13001
[39] [LR] Landi, G., Reina, C.: Symplectic dynamics on the universal Grassmannian and algebraic integrability. Preprint (1991)
[40] [Lep] Lepowsky, J.: Calculus of vertex operators. Proc. Natl. Acad. USA82, 8295–8299 (1985) · Zbl 0579.17010 · doi:10.1073/pnas.82.24.8295
[41] [Lu] Lu, J.H.: Poisson and symplectic structures of a Poisson Lie group. Preprint (December 1988)
[42] [LuW] Lu, J.H., Weinstein, A.: Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Diff. Geom.31, 501–526 (1990) · Zbl 0673.58018
[43] [McK] McKean, H.P.: Is there an infinite-dimensional algebraic geometry? Hints from KdV, Proc. Symp. in Pure Mathematics, vol.49, Providence, RI: Am. Math. Soc., 1989, pp. 27–37 · Zbl 0699.14053
[44] [Mi] Mikhalev, V.G.: A generalization of the Kac-Moody algebras with a parameter on an algebraic curve and perturbations of solitons. Commun. Math. Phys.134, 633–646 (1990) · Zbl 0757.35066 · doi:10.1007/BF02098450
[45] [vMM] van Moerbeke, P., Mumford, D.: The Spectrum of Difference Operators and Algebraic Curves. Acta Math.143, 93–154 (1979) · Zbl 0502.58032 · doi:10.1007/BF02392090
[46] [Mul1] Mulase, M.: Cohomological structure of soliton equations and Jacobian varieties. J. Diff. Geom.19, 403–430 (1984) · Zbl 0559.35076
[47] [Mul2] Mulase, M.: Category of vector bundles on algebraic curves and infinite dimensional Grassmannians. Int. J. Math.1, 293–342 (1990) · Zbl 0723.14010 · doi:10.1142/S0129167X90000174
[48] [Mum1] Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg de Vries equation and related non-linear equations. Intl. Symp. on Algebraic Geometry Kyoto, 1977, pp.115–153
[49] [Mum2] Mumford, D.: Tata lectures on theta, I, II. Boston: Birkhäuser 1983
[50] [PrS] Pressley, A., Segal, G.: Loop groups. Oxford: Clarendon Press 1986 · Zbl 0618.22011
[51] [PrW] Previato, E., Wilson, G.: Vectorbundles over curves and solutions of the KP equations. Theta functions, Bowdoin 1987, vol.49, Ehrenpreis, L., Gunning, R.C. (eds.) Proceedings of symposia in pure mathematics. Providence, RI: Am. Math. Soc., 1989, pp. 553–569
[52] [R] Read A.H.: A converse of Cauchy’s theorem and applications to extremal problems. Acta Math.100, 1–21 (1958) · Zbl 0142.04503 · doi:10.1007/BF02559600
[53] [RSTS] Reiman, A.G., Semenov-Tian-Shansky, M.A.: Reduction of Hamiltonian systems, Affine Lie algebras and Lax Equations I, II. Invent. Math.63, 423–432 (1981) · doi:10.1007/BF01389063
[54] [Ro] Royden, H.L.: The boundary values of analytic and harmonic functions. Math. Zeitschr78, 1–24 (1962) · Zbl 0196.33701 · doi:10.1007/BF01195147
[55] [Sa] Sato, M.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. RIMS Kokyoroku439, 30–46 (1981)
[56] [SeW] Segal, G.B., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. I.H.E.S.61, 5–65 (1985) · Zbl 0592.35112
[57] [STS1] Semenov-Tyan-Shanskii, M.A.: Poisson groups and dressing transformations. J. Sov. Math.46, 1641–1657 (1989) · Zbl 0673.35094 · doi:10.1007/BF01099196
[58] [STS2] Semenov-Tyan-Shanskii, M.A.: What is a classicalr-matrix? Funct. Anal. Appl.17(4), 259–272 (1983) · Zbl 0535.58031 · doi:10.1007/BF01076717
[59] [Ser] Serre, J.P.: Géométrie Algébrique et Géométrie Analytique. Collected Works, Volume 1, Berlin, Heidelberg, New York: Springer 1986, pp. 402–443
[60] [Sh] Shiota, T.: Characterization of Jacobians in terms of soliton equations. Inventiones83, 333–382 (1986) · Zbl 0621.35097 · doi:10.1007/BF01388967
[61] [Si] Simha, R.R.: The Behnke-Stein theorem for open Riemann surfaces. Proc. AMS105, 876 · Zbl 0674.30028
[62] [Sy] Symes, W.W.: Systems of Toda type, inverse spectral problems and representation theory. Inv. Math.59, 13–51 (1980) · Zbl 0474.58009 · doi:10.1007/BF01390312
[63] [UT] Ueno, K., Takasaki, K.: Toda Lattice Hierarchy. Advanced Studies in Mathematics4, 1–95 (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.