A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy

TL;DR

This paper demonstrates that the Ablowitz-Ladik hierarchy uniquely admits a full centerless Virasoro algebra of master symmetries, using CMV matrices to explicitly construct these symmetries and their Lax representation.

Contribution

It provides the first example of an integrable hierarchy with a complete centerless Virasoro algebra of master symmetries, expanding understanding of symmetries in integrable systems.

Findings

Ablowitz-Ladik hierarchy admits a centerless Virasoro algebra of master symmetries.

Explicit expressions for symmetries are given via generalized CMV matrices.

The hierarchy is the first known example with a full Virasoro algebra of master symmetries.

Abstract

We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Vel\'azquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the first example of an integrable hierarchy which admits a full centerless Virasoro algebra of master symmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchies which possess only "half of" a Virasoro algebra of master symmetries, as explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863-911], Damianou [Lett. Math. Phys. 20 (1990), 101-112] and Magri and Zubelli [Comm. Math. Phys. 141 (1991), 329-351].