Author: Sergei IGONIN
Following I. S. Krasilshchik and A. M. Vinogradov, we
regard PDEs as infinite-dimensional manifolds with involutive
distributions and consider their special morphisms called differential
coverings, which include constructions like Lax pairs and
Bæcklund transformations. We show that,
similarly to usual coverings in topology, at least for some PDEs
differential coverings are determined by actions of a sort of
fundamental group. This is not a group, but a certain
system of Lie algebras, which generalize Wahlquist-Estabrook algebras.
From this we deduce an algebraic necessary
condition for two PDEs to be connected by a Bæcklund transformation.
We compute these infinite-dimensional
Lie algebras for several KdV type equations
and prove non-existence of
As a by-product, for some class of Lie algebras g we prove that any subalgebra of g of finite codimension contains an ideal of g of finite codimension.
Revised version from May 12, 2005.
53 pages, LaTeX-2e (AmS-LaTeX).
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