# zbMATH — the first resource for mathematics

Solitons and infinite dimensional Lie algebras. (English) Zbl 0557.35091
Since 1981 the authors have published in collaboration with Date and Kashiwara a series of papers devoted to the analysis of the algebraic structure of soliton equations. These and the works of Drinfeld, Sokolov, Wilson and others constitute a recent surge on the study of soliton theory. Starting from a Clifford algebra A the authors give the definitions on the Fock representation of A, Lie algebras $$A_{\infty},B_{\infty},C_{\infty}$$ and $$D_{\infty}$$, which are respectively the infinite dimensional analogues of the classical Lie algebras $$A_ n$$, $$B_ n$$, $$C_ n$$ and $$D_ n$$. They discuss their reductions to the Kac-Moody Lie algebras $$A_ l^{(1)},C_ l^{(1)},D_ l^{(1)},A_ l^{(2)}$$ and $$D^{(2)}_{l+1}$$. The authors construct the so-called $$\tau$$-function in each case basing on the multiplicative group in the Clifford algebra and its Fock representation. These $$\tau$$-functions are shown to satisfy bilinear identities which can be rewritten into a series of nonlinear differential equations for $$\tau$$ (x). By introducing further transformations of dependent variables these equations turn out to be just soliton equations. The well-known soliton equations, such as the KP, KdV, MKdV, sine-Gordon, nonlinear Schrödinger, Heisenberg ferromagnet equations, the principal chiral model and many others including some difference equations can be generated in this way. This article contains also a number of useful tables such as the Chevalley basis for the above- mentioned Kac-Moody Lie algebras, a list of bilinear differential equations of low degree for the KP and modified KP hierarchies.
Reviewer: G.Tu

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 17B65 Infinite-dimensional Lie (super)algebras 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
Full Text:
##### References:
 [1] [ 2 ] Sato, M., RIMS Kokyuroku 439, Kyoto Univ. (1981) 30. [2] Kashiwara, M. and Miwa, T., Proc. Japan Acad. 57 A (1981) 342. [3] Date, E., Kashiwara, M. and Miwa, T., Proc. Japan Acad. 57 A (1981) 387. [4] Date, E., Jimbo, M., Kashiwara, M. and Miwa, T., /. Phys. Soc. Jpn. 50 (1981) 3806, 3813. [5] , Physica 4 D (1982) 343. [6] , Publ. RIMS, Kyoto Univ. 18 (1982) 1077, 1111. [7] , Transformation groups for soliton equations, Nonlinear Integrable Sys- tems - Classical Theory and Quantum Theory, Ed. by M. Jimbo and T. Miwa (World Scientific Publishing Company, Singapore, 1983). [8] , J.Phys. A 16 (1983) 221. [9] Miwa, T., Proc. Japan Acad. 58 A (1982) 9. [10] Date, E., Jimbo, M. and Miwa, T., /. Phys. Soc. Jpn. 51 (1982) 4116, 4125, 52 (1983) 388, 761, 766. [11] Jimbo, M. and Miwa, T., Lett. Math. Phys. 6 (1982) 463. [12] Jimbo, M. and Miwa, T., RIMS preprint 434, Kyoto Univ. (1983), submitted to Adv. Studies in Pure Math. [13] Ueno, K. and Nakamura, Y., Phys. Lett. 109 B (1982) 273. [14] , Phys. Lett. 117 B (1982) 208. [15] , to appear in Publ. RIMS, Kyoto Univ. 19 (1983). [16] Ueno, K., to appear in Publ RIMS, Kyoto Univ. 19 (1983). [17] Ueno, K. and Takasaki, Y., RIMS preprint 425, Kyoto Univ. (1983), submitted to Adv. Studies in Pure Math. [18] Drinfeld, V. G. and Sokolov, V. V., Dokl. Akad. Nauk USSR 258 (1981) 11. [19] Dolan, L., Phys. Rev. Lett. 47 (1981) 1371. [20] Chau, L. L., Ge, M. L. and Wu, Y. S., Phys. Rev. D 25 (1982) 1086. [21] Lepowsky, J. and Wilson, R. L., Comm. Math. Phys. 62 (1978) 43. [22] Kac. V. G., Kazhdan, D. A., Lepowsky, J. and Wilson, R. L., Adv. in Math. 42 (1981) 83. [23] Frenkel, I. B. and Kac, V. G., Inventions Math. 62 (1980) 23. [24] Kac, V. G. and Peterson, D. H., Proc. Nat’l. Acad. ScL, USA, 78 (1981) 3308. [25] Frenkel, I. B., Proc. Nat’l. Acad. Sci. USA, 77 (1980) 6303. [26] Sato, M., Miwa, T. and Jimbo, M., Publ. RIMS, Kyoto Univ. 14 (1978) 223, 15 (1979) 201, 577, 871, 16(1980) 531. [27] Jimbo, M., Miwa, T., M6ri, Y. and Sato, M., Physica I D (1980) 80. [28] Jimbo, M., Miwa, T. and Ueno, K., Physica 2 D (1981) 306,407, Physica 4 D (1981) 26. [29] Miwa, T., Publ. RIMS, Kyoto Univ. 17 (1981) 665. [30] Hirota, R., Direct Method in Soliton Theory, Solitons, Ed. by R. K. Bullough and P. J. Caudrey, Springer, 1980. [31] Littlewood, D. E., The theory of group characters, Oxford, 1950. · Zbl 0038.16504 [32] Zakharov, V. E. and Shabat, A. B., Funct. Anal, and Its Appl. 8 (1974) 226. [33] Satsuma, J., /. Phys. Soc. Jpn. 40 (1976) 286. [34] Gardner, C. S., Green, J. M., Kruskal, M. D. and Miura, R. M., Phys. Rev. Lett. 19 (1967) 1095. [35] Lax, P. D., Comm. Pure and Appl. Math. 21 (1968) 467. [36] Miura, R. M., /. Math. Phys. 9 (1968) 1202. [37] Hirota, R. and Satsuma, J., Progr. Theoret. Phys. 59, Supplement, (1976) 64. [38] Satsuma, J. and Ablowitz, M., /. Math. Phys. 20 (1979) 1496. [39] Zakharov, V. E. and Shabat, A. B., Sov. Phys. JETP 34 (1972) 62. [40] Nakamura, A. and Chen, H. H., /. Phys. Soc. Jpn. 49 (1980) 813. [41] Takhtajan, L. A., Phys. Lett. 64 (1977) 235. [42] Kaup, J., Stud, in Appl. Math. 62 (1980) 189. [43] Sawada, K. and Kotera, T., Progr. Theoret. Phys. 51 (1974) 1355. [44] Hirota, R. and Satsuma, J., Phys. Lett. 85 A (1981) 407. [45] , /. Phys. Soc. Jpn. 51 (1982) 3390. [46] Wilson, G., Phys. Lett. 89 A (1982) 332: see also Dodd, R. and Fordy, A., Phys. Lett. 89 A (1982) 168. [47] Ito, M., J. Phys. Soc. Jpn. 49(1980)771. [48] Toda, M., Hisenkei Kdshi Rikigaku, Iwanami, 1978; Theory of Nonlinear Lattices (English version), Springer, 1981. [49] Leznov, A. N. and Savelev, M. V., Lett. Math. Phys. 3 (1979) 489. [50] Hirota, R., /. Phys. Soc. Jpn. 51 (1982) 323. [51] Hirota, R. and Satsuma, J., /. Phys. Soc. Jpn. 40 (1976) 611. [52] Zakharov, V. E. and Mikhailov, A. V., Sov. Phys. JETP 47 (1978) 1017.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.