Abstract
Integrable hierarchies, viewed as isospectral deformations of an operatorL 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 onL, 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 ofL and an operatorM, 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 operatorà la Date-Jimbo-Kashiwara-Miwa, acting on the τ-function (or τ-vector). Lifting the algebra of symmetries, acting on wave functions, to an algebra of symmetries, acting on τ-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 τ-function (or τ-vectors).
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Communicated by A. Jaffe
The support of a National Science Foundation grant # DMS-9203246 is gratefully acknowledged.
The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.
The support of a National Science Foundation grant # DMS-9203246, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged.
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Adler, M., Shiota, T. & van Moerbeke, P. A Lax representation for the vertex operator and the central extension. Commun.Math. Phys. 171, 547–588 (1995). https://doi.org/10.1007/BF02104678
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DOI: https://doi.org/10.1007/BF02104678