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A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz–Ladik Hierarchy
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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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].
Keywords: Ablowitz–Ladik hierarchy; master symmetries; Virasoro algebra. 
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     author = {Luc Haine and Didier Vanderstichelen},
     title = {A {Centerless} {Virasoro} {Algebra} of {Master} {Symmetries} for the {Ablowitz{\textendash}Ladik} {Hierarchy}},
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Luc Haine; Didier Vanderstichelen. A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz–Ladik Hierarchy. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a78/

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