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

Symmetry Integrability and Geometry-Methods and Applications

article

Luc Haine¹  Didier Vanderstichelen¹ 

[1] Institut de Recherche en Mathématique et Physique, Université catholique de Louvain

关键词: Ablowitz–Ladik hierarchy;    master symmetries;    Virasoro algebra;   

DOI  :  10.3842/SIGMA.2013.079

来源: National Academy of Science of Ukraine

【 摘 要 】

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 Velázquez (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].

【 授权许可】

Unknown