## Generalized symmetries and $$w_\infty$$ algebras in three-dimensional Toda field theory.(English)Zbl 0972.81559

Summary: After establishing a formal theory for getting solutions of one type of high-dimensional partial differential equation, two sets of generalized symmetries of the 3D Toda theory, which arises from a particular reduction of the 4D self-dual gravity equation, are obtained concretely by a simple formula. Each set of symmetries constitutes a generalized $$w_\infty$$ algebra which contains three types of the usual $$w_\infty$$ algebras as special cases. Some open questions are discussed.

### MSC:

 81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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### References:

 [1] C. N. Pope, Phys. Lett. B 236 pp 173– (1990) [2] C. N. Pope, Nucl. Phys. B339 pp 191– (1990) [3] D. J. Gross, Phys. Rev. Lett. 64 pp 127– (1990) · Zbl 1050.81610 [4] M. Douglas, Nucl. Phys. B335 pp 635– (1990) [5] M. Douglas, Phys. Lett. B 238 pp 176– (1990) · Zbl 1332.81211 [6] E. Brezin, Phys. Lett. B 236 pp 144– (1990) [7] S-y Lou, Phys. Lett. B 302 pp 261– (1993) [8] S-y Lou, Phys. Lett. A 175 pp 23– (1993) [9] I. Bakas, Phys. Lett. B 228 pp 57– (1989) [10] A. Bilal, Phys. Lett. B 227 pp 406– (1989) [11] Q. H. Park, Nucl. Phys. B333 pp 267– (1990) [12] J. Avan, Phys. Lett. A 168 pp 363– (1992) [13] K. Yamagishi, Phys. Lett. B 259 pp 436– (1991) [14] A. Bial, Nucl. Phys. B326 pp 222– (1989) [15] V. Fateev, Nucl. Phys. B280 pp 644– (1987) [16] V. Fateev, Int. J. Mod. Phys. A 7 pp 853– (1991) · Zbl 0801.17017 [17] I. Bakas, Nucl. Phys. B343 pp 165– (1990) [18] M. Fukuma, Commun. Math. Phys. 143 pp 371– (1992) · Zbl 0757.35076 [19] D. David, J. Math. Phys. 27 pp 1225– (1986) · Zbl 0598.35117 [20] B. Fuchssteiner, Prog. Theor. Phys. 70 pp 1508– (1983) · Zbl 1098.37536 [21] B. Fuchssteiner, Prog. Theor. Phys. 65 pp 861– (1981) · Zbl 1074.58501 [22] B. Fuchssteiner, Nonlinear Analysis TMT 3 pp 849– (1979) · Zbl 0419.35049 [23] A. S. Fokas, Phys. Lett. 86A pp 341– (1981) [24] D. Olive, Nucl. Phys. B265 pp 469– (1986) [25] Q. H. Park, Phys. Lett. B 236 pp 423– (1990) [26] P. J. Olver, in: Applications of Lie Group to Differential Equations Berlin (1986) [27] I. I. Kogan, Mod. Phys. Lett. A 7 pp 3717– (1992) · Zbl 1021.81838 [28] K. M. Tamizhmani, J. Math. Phys. 32 pp 2635– (1991) · Zbl 0737.35110
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