On the variational cohomology of g-invariant foliations

Let S be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra g, we show that the 'vertical' variational cohomology of S is equal to the Lie algebra cohomology of g with values in the space of the 'horizontal' cohomology in a maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of infinitesimal actions leaving a given system invariant. (C) 2003 American Institute of Physics.44104702471

Invariants of pseudogroup actions: Homological methods and Finiteness theorem

We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of k-variants and k-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: We introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie-Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations.Comment: v2: some remarks and references addee

Variational Lie algebroids and homological evolutionary vector fields

We define Lie algebroids over infinite jet spaces and establish their equivalent representation through homological evolutionary vector fields.Comment: Int. Workshop "Nonlinear Physics: Theory and Experiment VI" (Gallipoli, Italy; June-July 2010). Published v3 = v2 minus typos, to appear in: Theoret. and Mathem. Phys. (2011) Vol.167:3 (168:1), 18 page

Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models

We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-type hyperbolic systems (including the 2D Toda chains), and Euler-Lagrange gauge theories (such as the Yang-Mills theories, gravity, or the Poisson sigma-models). Also, we formulate several open problems.Comment: Proc. 7th International Workshop "Quantum Theory and Symmetries-7" (August 7-13, 2011; CVUT Prague, Czech Republic), 20 page

THE COHOMOLOGY OF SYSTEMS OF DIFFERENTIAL-EQUATIONS AND LIE PSEUDOGROUPS

1045849

Group invariance of integrable Pfaffian systems

Let S be an integrable Pfaffian system. When it is invariant under a transversally free infinitesimal action of a finite-dimensional real Lie algebra g and consequently invariant under the local action of a Lie group G, we show that the vertical variational cohomology of S is equal to the Lie algebra cohomology of g with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant19851863188

JORDAN-HOLDER SERIES AND PRINCIPLES OF A LIE GROUP

15330735