Abstract
The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of WKP admitting a central extension for generic values of the parameter, reducing naturally to W n for special values of the parameter, and contracting to the centrally extended W1+∞, W∞ and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic tow KP. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of\(\hat W_\infty\) which contracts to a new nonlinear algebra of the W∞-type.
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References
Bouwknegt, P., Schoutens, K.:W-Symmetry in Conformal Field Theory. Phys. Reps., to appear
Figueroa-O'Farrill, J.M., Ramos, E.: J. Math. Phys.33, 833 (1992)
Zamolodchikov, A.B.: Theor. Mat. Phys.65, 1205 (1986); Fateev, V.A., Lykyanov, S.L.: Int. J. Mod. Phys.A3, 507 (1988)
Gel'fand, I.M., Dickey, L.A.: A family of Hamiltonian structures connected with integrable nonlinear differential equations. Preprint 136, IPM AN SSSR, Moscow (1978)
Dickey, L.A.: Soliton equations and Hamiltonian systems. Advanced Series in Mathematical Physics Vol.12, Singapore: World Scientific Publ. Co.
Adler, M.: Invent. Math.50, 403 (1979)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Proc. Jpn. Acad. Sci.57A, 387 (1981); J. Phys. Soc. Jpn.50, 3866 (1981)
Dickey, L.A.: Annals NY Acad. Sci.491, 131 (1987); Figueroa-O'Farrill, J.M., Mas, J., Ramos, E.: Phys. Lett266B, 298 (1991); Yu, F., Wu, Y.-S.: Nucl. Phys.B373, 173 (1992)
Figueroa-O'Farrill, J.M., Ramos, E.: Phys. Lett.282B, 357 (1992) (hep-th/9202040); ClassicalW-algebras from dispersionless Lax hierarchies. Preprint, to appear
Bakas, I., Kiritsis, E.: Maryland/Berkeley/LBL preprint UCB-PTH-P1/44, LBL-31213 and UMD-PP-92/37, Sept. 1991
Yu, F., Wu, Y.-S.:UU-HEP-92/11 (hep-th/9205117)
Yu, F., Wu, Y.-S.: Phys. Rev. Lett.68, 2996 (1992) (hep-th/9112009)
Radul, A.O.: In Applied methods of nonlinear analysis and control. pp. 149–157, Mironov, Moroz, Tshernjatin, (eds.) MGU. 1987 (Russian)
Das, A., Huang, W.-J.: J. Math. Phys.33, 2487 (1992)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Berlin, Heidelberg, New York: Springer 1987
Bakas, I.: Phys. Lett.228B, 406 (1989); Commun. Math. Phys.134, 487 (1990)
Pope, C.N., Romans, L.J., Shen, X.: Nucl. Phys.B339, 191 (1990)
Pope, C.N., Romans, L.J., Shen, X.: Phys. Lett.242B, 401 (1990)
Yamagishi, K.: Phys. Lett.259B, 436 (1991); Yu, F., Wu, Y.-S.: Phys. Lett.236B, 220 (1991)
Watanabe, Y.: Lett. Math. Phys.7, 99 (1983); Annali di Matematica136, 77 (1984)
Bakas, I., Khesin, B., Kiritsis, E.: UCB-PTH-91/48, LBL-31303, UMD-PP-92-36
Depireux, D.: Preprint LAVAL PHY-21-92, hep-th/9203062
Drinfel'd, V.G., Sokolov, V.V.: J. Soviet Math.30, 1975 (1984)
Aratyn, H., Ferreira, L.A., Gomes, J.F., Zimerman, A.H.: Preprint IFT-P/020/92-SAO-PAULO, hep-th/9206096
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Communicated by N. Yu. Reshetikhin
Address after October 1993: Queen Mary and Westfield College, UK
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Figueroa-O'Farrill, J.M., Mas, J. & Ramos, E. A one-parameter family of hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinear WKP algebra. Commun.Math. Phys. 158, 17–43 (1993). https://doi.org/10.1007/BF02097230
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DOI: https://doi.org/10.1007/BF02097230