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A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2013, том 9
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A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
Інший ID:
2010 Mathematics Subject Classification: 37K10; 17B68
DOI: http://dx.doi.org/10.3842/SIGMA.2013.079
DOI: http://dx.doi.org/10.3842/SIGMA.2013.079
Посилання:
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy / L. Haine, D. Vanderstichelen // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 39 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full
collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html.
The authors thank the referees for their useful comments on this work and for drawing attention to the references [27, 28, 29]. The first author acknowledges the partial support of the
Belgian Interuniversity Attraction Poles P06/02 and P07/18. During part of this research, the
second author was a Research Fellow of the Belgian National Science Foundation (FNRS), whose
support is also gratefully acknowledged.
Дата:
2013
Переглядів:
770
Завантажень:
291
Анотація:
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].
