Igonin S. Lie algebras responsible for ZCRs of (1+1)-dimensional PDEs and some applications to Bäcklund transformations, talk at The Workshop on Geometry of PDEs and Integrability, 14-18 October 2013, Teplice nad Becvou, Czech Republic (abstract)

Speaker: Sergei Igonin

Title: Lie algebras responsible for zero-curvature representations of (1+1)-dimensional PDEs and some applications to Bäcklund transformations

Abstract:
For any PDE satisfying some non-degeneracy conditions, we can define a family of Lie algebras, which are called the fundamental Lie algebras of this PDE. Fundamental Lie algebras are defined in a coordinate-independent way and are new geometric invariants for PDEs.

Recall that, for every topological space $X$ and every point $a\in X$, one has the fundamental group $\pi _{1}(X,a)$. The above-mentioned Lie algebras are called fundamental, because their role for PDEs is somewhat similar to the role of fundamental groups for topological spaces.

Fundamental Lie algebras are closely related to zero-curvature representations and Bäcklund transformations. In this talk, we will concentrate on the case of analytic (1+1)-dimensional (multicomponent) evolution PDEs. For such PDEs, it will be shown how fundamental Lie algebras classify all zero-curvature representations up to local gauge equivalence. Also, we will show how to describe fundamental Lie algebras in terms of generators and relations. Using these algebras, one obtains necessary conditions for integrability and necessary conditions for existence of Bäcklund transformations for the considered class of PDEs.

In our construction, jets of arbitrary order are allowed. In the case of low-order jets, fundamental Lie algebras generalize Wahlquist-Estabrook prolongation algebras.

In the structure of fundamental Lie algebras for KdV, Krichever-Novikov, nonlinear Schrödinger, (multicomponent) Landau-Lifshitz type equations, we encounter infinite-dimensional subalgebras of Kac-Moody algebras and infinite-dimensional Lie algebras of certain matrix-valued functions on some algebraic curves. Applications to classification of some evolution PDEs with respect to Bäcklund transformations will be discussed as well.

Slides: Igonin S. Lie algebras responsible for zero-curvature representations of (1+1)-dimensional PDEs and some applications to Bäcklund transformations (present Workshop Geom. of PDEs Integr., 14-18 Oct 2013, Teplice nad Becvou, Czech Rep).pdf