Igonin S. Lie algebras associated with PDEs and Bäcklund transformations, talk at Conf. The Interface of Integrability and Quantization, Lorentz Center, Leiden (The Netherlands), 2010 (abstract): Difference between revisions
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| speaker = | | speaker = Sergei Igonin | ||
| title = Lie algebras associated with PDEs and Bäcklund transformations | | title = Lie algebras associated with PDEs and Bäcklund transformations | ||
| abstract = We introduce a new geometric invariant of PDEs: with any analytic system of PDEs we associate naturally a certain system of Lie algebras. These Lie algebras are responsible for Bäcklund transformations (a tool to construct exact solutions for nonlinear PDEs) and zero-curvature representations (including 2-dimensional Lax pairs) in the theory of integrable systems. Using infinite jet spaces, we regard PDEs as infinite-dimensional manifolds with involutive distributions and study their special morphisms called Krasil'shchik-Vinogradov coverings, which generalize the classical concept of coverings from topology and provide a geometric framework for Bäcklund transformations, Lax pairs, and some other constructions in soliton theory. Recall that topological coverings of a manifold M can be described in terms of the fundamental group of M. We show that a similar description exists for finite-rank Krasilshchik-Vinogradov coverings of PDEs. | | abstract = We introduce a new geometric invariant of PDEs: with any analytic system of PDEs we associate naturally a certain system of Lie algebras. These Lie algebras are responsible for Bäcklund transformations (a tool to construct exact solutions for nonlinear PDEs) and zero-curvature representations (including 2-dimensional Lax pairs) in the theory of integrable systems. Using infinite jet spaces, we regard PDEs as infinite-dimensional manifolds with involutive distributions and study their special morphisms called Krasil'shchik-Vinogradov coverings, which generalize the classical concept of coverings from topology and provide a geometric framework for Bäcklund transformations, Lax pairs, and some other constructions in soliton theory. Recall that topological coverings of a manifold M can be described in terms of the fundamental group of M. We show that a similar description exists for finite-rank Krasilshchik-Vinogradov coverings of PDEs. | ||
However, the "fundamental group of a PDE" is not a group, but a certain system of Lie algebras, which we call fundamental algebras. We have computed these algebras for a number of well-known nonlinear PDEs. As a result, one obtains infinite-dimensional Lie algebras of Kac-Moody type and Lie algebras of matrix-valued functions on algebraic curves. Applications to construction and classification of Bäcklund transformations will be also presented. | However, the "fundamental group of a PDE" is not a group, but a certain system of Lie algebras, which we call fundamental algebras. We have computed these algebras for a number of well-known nonlinear PDEs. As a result, one obtains infinite-dimensional Lie algebras of Kac-Moody type and Lie algebras of matrix-valued functions on algebraic curves. Applications to construction and classification of Bäcklund transformations will be also presented. | ||
| slides = | | slides = [[Media:Igonin_S._Lie_algebras_associated_with_PDEs_and_Backlund_transformations_%28presentation_at_the_Lorentz_Center_Workshop_The_Interface_of_Integrability_and_Quantization%2C_Leiden%2C_The_Netherlands%2C_12-16_April_2010%29.pdf|Igonin S. Lie algebras associated with PDEs and Backlund transformations (presentation at the Lorentz Center Workshop The Interface of Integrability and Quantization, Leiden, The Netherlands, 12-16 April 2010).pdf]] | ||
| references = | | references = | ||
| 79YY-MM-DD = 7989-95-84 | | 79YY-MM-DD = 7989-95-84 | ||
}} | }} | ||
Latest revision as of 23:06, 15 January 2014
Speaker: Sergei Igonin
Title: Lie algebras associated with PDEs and Bäcklund transformations
Abstract:
We introduce a new geometric invariant of PDEs: with any analytic system of PDEs we associate naturally a certain system of Lie algebras. These Lie algebras are responsible for Bäcklund transformations (a tool to construct exact solutions for nonlinear PDEs) and zero-curvature representations (including 2-dimensional Lax pairs) in the theory of integrable systems. Using infinite jet spaces, we regard PDEs as infinite-dimensional manifolds with involutive distributions and study their special morphisms called Krasil'shchik-Vinogradov coverings, which generalize the classical concept of coverings from topology and provide a geometric framework for Bäcklund transformations, Lax pairs, and some other constructions in soliton theory. Recall that topological coverings of a manifold M can be described in terms of the fundamental group of M. We show that a similar description exists for finite-rank Krasilshchik-Vinogradov coverings of PDEs.
However, the "fundamental group of a PDE" is not a group, but a certain system of Lie algebras, which we call fundamental algebras. We have computed these algebras for a number of well-known nonlinear PDEs. As a result, one obtains infinite-dimensional Lie algebras of Kac-Moody type and Lie algebras of matrix-valued functions on algebraic curves. Applications to construction and classification of Bäcklund transformations will be also presented.