Additional Symmetries of Generalized Hierarchies

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 rôle 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.

24 pages, (macro included), CERN-TH.6998/93, US-FT-8/93, SWAT/93-92/10

References in corpus (6)

• Tau-Functions and Generalized Integrable Hierarchies
• Additional symmetries of KP, Grassmannian, and the string equation II
• Generalized W-algebras and Integrable Hierarchies
• The Lax Operator Approach for the Virasoro and the W-Constraints in the Generalized KdV Hierarchy
• Generalized Integrability and two-dimensional Gravitation
• New Topological Theories and Conjugacy Classes of the Weyl Group

Cited by in corpus (8)

• Virasoro Symmetry of Constrained KP Hierarchies
• Tau function of the CKP hierarchy and non-linearizable Virasoro symmetries
• Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies
• Symmetries of Supersymmetric Integrable Hierarchies of KP Type
• Simple Lie algebras, Drinfeld--Sokolov hierarchies, and multi-point correlation functions
• Tau Functions and Virasoro Symmetries for Drinfeld-Sokolov Hierarchies
• Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials
• The KdV vacuum operator and its Katz extension