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A Lax representation for the vertex operator and the central extension. (English) Zbl 0839.35116
Summary: Integrable hierarchies, viewed as isospectral deformations of an operator $$L$$ may admit symmetries; they are time-dependent vector fields, transversal to and commuting with the hierarchy and forming an algebra. In this work, the commutation relations for the symmetries are shown to be based on a non-commutative Lie algebra splitting theorem. The symmetries, viewed as vector fields on $$L$$, are expressed in terms of a Lax pair.
This study introduces a “generating symmetry”, a generating function for symmetries, both of the KP equation (continuous), and the two-dimensional Toda lattice (discrete), in terms of $$L$$ and an operator $$M$$, introduced by Orlov and Schulman, such that $$[L, M]= 1$$. This “generating symmetry”, acting on the wave function (or wave vector) lifts to a vertex operator à la Date-Jimbo-Kashiwara-Miwa, acting on the $$\tau$$-function (or $$\tau$$-vector). Lifting the algebra of symmetries, acting on wave functions, to an algebra of symmetries, acting on $$\tau$$-functions, amounts to passing from an algebra to its central extension.
This provides a handy technology to find the constraints satisfied by various matrix integrals, arising in the context of $$2d$$-quantum gravity and moduli space topology. The point is to first prove the vanishing of symmetries at the Lax pair level, which usually turns out to be elementary and conceptual, and then use the lifting above to get the subalgebra of vanishing symmetries for the $$\tau$$-function (or $$\tau$$-vectors).

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 58J70 Invariance and symmetry properties for PDEs on manifolds 17B66 Lie algebras of vector fields and related (super) algebras
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##### References:
 [1] [A-vM0] Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math.38, 267–317 (1980) · Zbl 0455.58017 · doi:10.1016/0001-8708(80)90007-9 [2] [A-vM1] Adler, M., van Moerbeke, P.: A matrix integral solution to two-dimensionalW p -Gravity. Commun. Math. Phys.147, 25–56 (1992) · Zbl 0756.35074 · doi:10.1007/BF02099527 [3] [A-vM2] Adler, M., van Moerbeke, P.: Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials. To appear in Duke Math J. (1995) · Zbl 0848.17027 [4] [A-vm3] Adler, M., van Moerbeke, P.: Two-matrix integrals and two-Toda symmetries. To appear · Zbl 0848.17027 [5] [A-Sh-vM] Adler, M., Shiota, T., van Moerbeke, P.: From thew algebra to its central extension: a $$\tau$$-function approach. To appear in Physics Letters A. (1994) · Zbl 0925.58031 [6] [A-K] Aoyama, S., Kodama, Y.: A generalized Sato equation and theW algebra. Phys. Lett.B278 56–62 (1992) · doi:10.1016/0370-2693(92)90711-C [7] [BX] Bonora, L., Xiong, C.S.: Matrix models without scaling limit. SISSA reprint [8] [B-K] Brézin, E., Kazakov, V.A.: Exactly solvable field theories of closed strings. Phys. Lett.236B, 144–150 (1990) [9] [DJKM] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Proc. RIMS Symp. Nonlinear integrable systems-Classical theory and quantum theory (Kyoto 1981), Singapore: World Scientific, 1983, pp. 39–119 [10] [D1] Dickey, L.: Soliton equations and integrable systems. Singapore. World Scientific, 1991 [11] [D2] Dickey, L.: Lectures on classicalW-algebras. Cortona, Italy 1993 [12] [D3] Dickey, L.A.: On additional symmetries of the KP hierarchy and Sato’s Bäcklund transformation. Commun. Math. Phys. · Zbl 0813.35106 [13] [DG] Duistermaat, J.J., Grünbaum, F.A.: Differential equations in the spectral parameter. Commun. Math. Phys.103, 177–240 (1986) · Zbl 0625.34007 · doi:10.1007/BF01206937 [14] [Fa] Fastré, J.: Boson-correspondence for W-algebras, Bäcklund-Darboux transformations and the equation [L, P]=L n . University of Louvain, doctoral dissertation (1993) [15] [Fu] Fuchssteiner, B.: Mastersymmetries, Higher Order Time-Dependent Symmetries and Conserved Densities of Nonlinear Evolution Equations. Prog. Theor. Phys.70, No. 6, 1508–1522 (1983) · Zbl 1098.37536 · doi:10.1143/PTP.70.1508 [16] [FKN] Fukuma, M., Kawai, H., Nakayama, R.: Infinite dimensional Grassmannian structure of two-dimensional Quantum gravity. Commun. Math. Phys.143, 371–403 (1992) · Zbl 0757.35076 · doi:10.1007/BF02099014 [17] [GMO] Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., Orlov, A.: Matrix models of Two-dimensional gravity and Toda theory. Nucl. Phys.B357, 565–618 (1991) · doi:10.1016/0550-3213(91)90482-D [18] [Gr-OS] Grinevich, P.G., Orlov, A.Yu., Schulman, E.I.: On the Symmetries of the integrable system. In: Modern development of the Soliton theory, ed. A. Fokas, V.E. Zakharov, (1992) [19] [Gr-O] Grinevich, P.G., Orlov, A.Yu.: Flag spaces in KP theory and Virasoro action on DetD j and Segal-Wilson tau-function. In: Modern problems of Quantum Field theory, Belavin, Zamolodchikov, Klimuk (eds.) Berlin, Heidelberg, New York: Springer, 1989 [20] [Ka] Kac, V.G.: Infinite-dimensional Lie algebras. 3rd edition, Cambridge: Cambridge Univ. Press, 1990 · Zbl 0716.17022 [21] [K-R] Kac, V.G., Raina, A.K.: Bombay lectures on highest weight representations of infinite dimensional Lie Algebras. Adv. Series Math. Phys. vol. 2, 1987 · Zbl 0668.17012 [22] [K-S] Kac, V.G., Schwarz, A.: Geometric interpretation of partition function of 2D gravity. Phys. Lett.257B, 329–334 (1991) [23] [K] Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys.147, 1–23 (1992) · Zbl 0756.35081 · doi:10.1007/BF02099526 [24] [M-Z] Magri, F., Zubelli, J.P.: Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV. Commun. Math. Phys.141 (2), 329–352, (1991) · Zbl 0743.35072 · doi:10.1007/BF02101509 [25] [Me] Mehta, M.L.: A method of integration over matrix variables. Commun. Math. Phys.79, 327–340 (1981) · Zbl 0471.28007 · doi:10.1007/BF01208498 [26] [N] Niedermaier, M.:W 1+$$\alpha$$ as a symmetry ofGL $$\alpha$$-orbits, preprint 1993 [27] [O-Fa] Oevel, W., Falck, M.: Master Symmetries for Finite-Dimensional Integrable Systems: the Calogero-Moser system. Prog. Theor. Phys.75, 1328–1341 (1986) · doi:10.1143/PTP.75.1328 [28] [O-Fu] Oevel, W., Fuchssteiner, B.: Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation. Phys. Lett.88A, 323–327 (1982) [29] [O-F-B] Oevel, G., Fuchssteiner, B., Blaszak, M.: Action-Angle Representation of Multisolitons by Potentials of Mastersymmetries. Prog. Theo. Phys.83, N0 3, 395–413 (1990) · Zbl 1058.37527 · doi:10.1143/PTP.83.395 [30] [O] Orlov, A.Y.: Vertex operator, $$\bar \partial$$ -problem, symmetries, variational identities and Hamiltonian formalism for 2+1 integrable systems. In: Proc. Kiev Intern. Workshop ”Plasma Theory and Non-linear and Turbulent Processes in Physics”, Baryakhtar (ed.) Singapore: World Scientific, 1988 [31] [O-Sc] Orlov, A.Y., Schulman, E.I.: Additional Symmetries for Integrable and Conformal Algebra Representation. Lett. Math. Phys.12, 171–179 (1986) · Zbl 0618.35107 · doi:10.1007/BF00416506 [32] [Sa1] Sato, M.: Soliton equations and the universal Grassmann manifold (by Noumi in Japanese). Math. Lect. Note Ser. No.18, Sophia University, Tokyo, 1984 · Zbl 0541.58001 [33] [Sa2] Sato, M.: The KP Hierarchy and Infinite-Dimensional Grassmann Manifolds. Proceedings of Symposia in Pure Mathematics, Vol.149 Part 1, (1989) pp. 51–66 [34] [S-S] Sato, M., and Sato, Y.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. Lect. notes in Num. Appl. Anal. Vol.5, 259–271 (1982) [35] [Sc] Schwarz, A.: On solutions to the string equation. Mod. Phys. Lett.A 6, 2713–2725 (1991) · Zbl 1020.37579 · doi:10.1142/S0217732391003171 [36] [Sh] Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations. Invent. Math.83, 333–382 (1986) · Zbl 0621.35097 · doi:10.1007/BF01388967 [37] [T-T] Takasaki, K., Takebe, T.: Integrable hierarchies and dispersionless limit. Reviews in Mathematical Physics, to appear · Zbl 0838.35117 [38] [U-T] Ueno, K., Takasaki, K.: Toda Lattice Hierarchy. Adv. Studies in Pure Math., Vol.4, 1–95 (1984) [39] [v1] van de Leur, J.: TheW 1+$$\alpha$$(gl s )-symmetries of thes-component KP-hierarchy, preprint 1994 [40] [v2] van de Leur, J.: The Adler-Shiota-van Moerbeke formula for the BKP-hierarchy, preprint 1994 · Zbl 0844.35109 [41] [vM-M] 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 [42] [vM] van Moerbeke, P.: Integrable foundations of string theory. CIMPA Summer School at Sophia-Antipolis (June 1991), Lectures on Integrable systems, Ed.: O. Babelon, P. Cartier, Y. Kosmann-Schwarzbach, World Scientific, 1994, pp. 163–267 [43] [W2] Witten, E.: On the Kontsevich Model and other Models of Two Dimensional Gravity. IASSNS-HEP-91/24 (6/1991), Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol.1,2 (New York, 1991), 176–216, World Sci. Publishing, River Edge, NJ 1992
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