Additional Symmetries of Generalized Hierarchies
T.J. Hollowood, J.L. Miramontes, J. Sanchez Guillen

TL;DR
This paper identifies and analyzes non-isospectral symmetries in generalized integrable hierarchies, revealing their algebraic structure and implications for tau-functions and quantum gravity models.
Contribution
It generalizes known symmetries of integrable hierarchies using Virasoro and Kac--Moody algebra structures, expanding understanding of their algebraic properties.
Findings
Symmetries form a subalgebra of the Virasoro algebra.
Infinitesimal symmetries act on tau-functions as Virasoro currents.
Comments on implications for matrix-model quantum gravity.
Abstract
The non-isospectral symmetries of a general class of integrable hierarchies are found, generalizing the Galilean and scaling symmetries of the Korteweg--de Vries equation and its hierarchy. The symmetries arise in a very natural way from the semi-direct product structure of the Virasoro algebra and the affine Kac--Moody algebra underlying the construction of the hierarchy. In particular, the generators of the symmetries are shown to satisfy a subalgebra of the Virasoro algebra. When a tau-function formalism is available, the infinitesimal symmetries act directly on the tau-functions as moments of Virasoro currents. Some comments are made regarding the r\^ole of the non-isospectral symmetries and the form of the string equations in matrix-model formulations of quantum gravity in two-dimensions and related systems.
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