A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
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,...
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| Zitieren: | 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. |
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Haine, L. Vanderstichelen, D. 2019-02-21T07:26:52Z 2019-02-21T07:26:52Z 2013 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. 1815-0659 2010 Mathematics Subject Classification: 37K10; 17B68 DOI: http://dx.doi.org/10.3842/SIGMA.2013.079 https://nasplib.isofts.kiev.ua/handle/123456789/149371 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]. 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy |
| spellingShingle |
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy Haine, L. Vanderstichelen, D. |
| title_short |
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy |
| title_full |
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy |
| title_fullStr |
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy |
| title_full_unstemmed |
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy |
| title_sort |
centerless virasoro algebra of master symmetries for the ablowitz-ladik hierarchy |
| author |
Haine, L. Vanderstichelen, D. |
| author_facet |
Haine, L. Vanderstichelen, D. |
| publishDate |
2013 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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].
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149371 |
| citation_txt |
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. |
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2025-12-07T20:31:44Z |
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