A Lax representation for the vertex operator and the central extension
A Lax representation for the vertex operator and the central extension
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DOI：
10.1007/bf02104678
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发表时间：
1995-08
影响因子：
2.4
通讯作者：
M. Adler;T. Shiota;P. Moerbeke
中科院分区：
文献类型：
--
作者：
M. Adler;T. Shiota;P. Moerbeke
Integrable hierarchies, viewed as isospectral deformations of an operatorLmay 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 ofLand 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 laDate-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).
