Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras
We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the Kadomtsev-Petviashvili (KP) hierarchy, together with its generalizations and reduction to the Nth Korteweg-de Vries (KdV) hierarchy, us...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
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Oxford University Press (OUP)
2018
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| Online Access: | http://hdl.handle.net/1721.1/115854 https://orcid.org/0000-0002-2860-7811 |
| _version_ | 1826193811374604288 |
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| author | De Sole, Alberto Valeri, Daniele Kac, Victor |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics De Sole, Alberto Valeri, Daniele Kac, Victor |
| author_sort | De Sole, Alberto |
| collection | MIT |
| description | We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the Kadomtsev-Petviashvili (KP) hierarchy, together with its generalizations and reduction to the Nth Korteweg-de Vries (KdV) hierarchy, using the formal distribution calculus and the λ-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (nonlocal) bi-Poisson structures of the matrix KP and the matrix Nth KdV hierarchies, and we prove integrability of the Nth matrix KdV hierarchy. |
| first_indexed | 2024-09-23T09:45:47Z |
| format | Article |
| id | mit-1721.1/115854 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T09:45:47Z |
| publishDate | 2018 |
| publisher | Oxford University Press (OUP) |
| record_format | dspace |
| spelling | mit-1721.1/1158542022-09-30T16:40:02Z Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras De Sole, Alberto Valeri, Daniele Kac, Victor Massachusetts Institute of Technology. Department of Mathematics Kac, Victor We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the Kadomtsev-Petviashvili (KP) hierarchy, together with its generalizations and reduction to the Nth Korteweg-de Vries (KdV) hierarchy, using the formal distribution calculus and the λ-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (nonlocal) bi-Poisson structures of the matrix KP and the matrix Nth KdV hierarchies, and we prove integrability of the Nth matrix KdV hierarchy. 2018-05-24T17:30:54Z 2018-05-24T17:30:54Z 2015-02 2014-01 2018-05-23T18:07:54Z Article http://purl.org/eprint/type/JournalArticle 1073-7928 1687-0247 http://hdl.handle.net/1721.1/115854 De Sole, Alberto et al. “Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras.” International Mathematics Research Notices 2015, 21 (February 2015): 11186–11235 © 2015 The Author(s) https://orcid.org/0000-0002-2860-7811 http://dx.doi.org/10.1093/IMRN/RNV017 International Mathematics Research Notices Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press (OUP) arXiv |
| spellingShingle | De Sole, Alberto Valeri, Daniele Kac, Victor Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras |
| title | Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras |
| title_full | Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras |
| title_fullStr | Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras |
| title_full_unstemmed | Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras |
| title_short | Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras |
| title_sort | adler gelfand dickey approach to classical 𝒲 algebras within the theory of poisson vertex algebras |
| url | http://hdl.handle.net/1721.1/115854 https://orcid.org/0000-0002-2860-7811 |
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