Geodesic
On a $c$-number quantum $\tau $-function
Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 1, pp. 119-131 Cet article a éte moissonné depuis la source Math-Net.Ru

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We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $\tau$-function nor the one associated with non-Cartanian (level $k\ne 1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields. 
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A. D. Mironov; A. Yu. Morozov; L. Vinet. On a $c$-number quantum $\tau $-function. Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 1, pp. 119-131. http://geodesic.mathdoc.fr/item/TMF_1994_100_1_a11/

[1] M. Sato, Y. Sato, “Soliton equations as dynamical systems in an infinite-dimensional Grassmannian”, Nonlinear partial differential equations in applied sciences, eds. P. Lax, H. Fujita, G. Strang, North-Holland, Amsterdam, 1982 ; E. Date, M. Jimbo, M. Kashiwara, T. Miwa, “Transformation groups for soliton equations”, Symp. “Non-linear integrable systems – classical theory and quantum theory” (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39–119 ; K. Ueno, K. Takasaki, “Toda lattice hierarchy”, Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., 4, North-Holland, Amsterdam, 1984, 1–95 | MR | MR | MR

[2] G. Segal, G. Wilson, Publ. Math. IHES, 61 (1985), 1 | DOI

[3] V. Kac, Infinite-dimensional Lie algebras, chapter 14, Cambridge University Press, Cambridge, 1985 ; V. Kac, J. van der Leur, “The $n$-component KP hierarchy and representation theory”, Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993, 302–343 | MR | Zbl | DOI | MR | Zbl

[4] E. Brézin, V. Kazakov, Phys. Lett. B, 236 (1990), 144 | DOI | MR

[5] A. Morozov, Soviet Phys. Uspekhi, 29:11 (1986), 993–1039 | DOI | MR

[6] K. Kajiwara, J. Satsuma, J. Phys. Soc. Jpn., 60 (1991), 3986 ; K. Kajiwara, Y. Ohta, J. Satsuma, “$q$-Discrete Toda Molecule Equation”, Phys. Lett. A, 180:3 (1993), 249–256 | DOI | MR | DOI | MR

[7] F. Smirnov, “What are we quantizing in integrable field theory”, Algebra i Analiz, 6:2 (1994), 248–261 ; B. Feigin, E. Frenkel, Integrals of Motion and Quantum groups, preprint, september 1993 | MR | Zbl | MR

[8] A. Marshakov, Int. J. Mod. Phys. A, 8 (1993), 3831 ; A. Mironov, $2d$ gravity and matrix models. I. 2d gravity, preprint UBC/S-95/93, FIAN/TD-24/93, august 1993 | DOI | MR | Zbl | MR

[9] S. Kharchev et al., Generalized Kazakov–Migdal–Kontsevich Model: group theory aspects, preprint UUITP-10/93, FIAN/TD-07/93, ITEP-M4/93 | MR

[10] R. Hirota, J. Phys. Soc. Japan., 50 (1981), 3785 ; T. Miwa, Proc. Japan. Acad. A, 58 (1982), 8 ; S. Saito, Phys. Rev. Lett., 59 (1987), 1798 ; Phys. Rev. D, 36 (1987), 1819 ; S. Kharchev, A. Mironov, $\tau$-function of two-dimensional Toda lattice and quantum deformations of the integrable hierarchies, preprint FIAN/TD-02/93 | DOI | MR | MR | DOI | MR | DOI | MR

[11] S. Kharchev et al., Nucl. Phys. B, 366 (1991), 569 | DOI | MR

[12] A. Morozov, L. Vinet, Mod. Phys. Lett. A, 8 (1993), 2891 | DOI | MR | Zbl

[13] G. Lusztig, Adv. Math., 70 (1988), 237 | DOI | MR | Zbl

[14] V. Drinfeld, Sov. Math. Dokl., 32 (1985), 254; 36 (1987), 212 ; M. Jimbo, Lett. Math. Phys., 10 (1985), 63 ; I. Frenkel, N. Jing, Proc. Nat. Acad. Sci. USA, 85 (1988), 9373 | MR | DOI | MR | Zbl | DOI | MR | Zbl