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The geometry of spinors and the multicomponent BKP and DKP hierarchies. (English) Zbl 0924.35114

Harnad, John (ed.) et al., The bispectral problem. Proceedings of the CRM workshop, Montréal, Canada, March 1997. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 14, 159-202 (1998).
In [M. Sato, Proc. Symp., Kyoto 1981, RIMS Kokyuroku 439, 30-46 (1981; Zbl 0507.58029), cf. also V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Adv. Ser. Math. Phys., Vol. 2. Singapore-New Jersey-Hong Kong: World Scientific. IX (1987; Zbl 0668.17012)], the Hirota equations for the KP hierarchy are shown to be equivalent to a set of Plücker relations satisfied on the \(GL_\infty\) orbit of the vacuum tau function. This paper extends the theory to the BKP hierarchy by the introduction of the formalism of spinor geometry. The multicomponent KP is obtained as a reduction. Both the Sato and Lax equations are derived, as well as a dictionary that translates the operation of a \(W\) algebra on the tau function into additional symmetries of the wave function [A. Yu. Orlov and E. I. Shul’man, Lett. Math. Phys. 12, 171-179 (1986; Zbl 0618.35107); M. Adler, T. Shiota and P. van Moerbeke, Commun. Math. Phys. 171, No. 3, 547-588 (1995; Zbl 0839.35116) and J. van de Leur, J. Math. Phys. 37, No. 5, 2315-2337 (1996; Zbl 0864.35096)]. The authors note that different bosonizations produce different hierarchies of PDEs.
For the entire collection see [Zbl 0885.00050].

MSC:

35Q40 PDEs in connection with quantum mechanics
81R25 Spinor and twistor methods applied to problems in quantum theory
17B65 Infinite-dimensional Lie (super)algebras
17B68 Virasoro and related algebras
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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