Invariants differentiels d'un pseudogroupe de Lie

Recent Advances in the Theory and Application of Lie Pseudo-Groups

“…Clearly, any functional combination of differential invariants is a differential invariant (on the common domain of definition) and thus we speak, somewhat loosely, of the algebra of differential invariants associated with the action of the transformation group on submanifolds of a specified dimension. Since differential invariants may only be locally defined, we should technically work the category of sheaves of differential invariants, [40,41]. However, for our local constructions, this extra level of abstraction only serves to obscure the main ideas, and so we will leave their sheaf-theoretic reformulation as a simple translational exercise for the cognoscenti.…”

“…Thus, freeness serves to bound the number of pseudo-group parameters at each sufficiently high order, providing a simpler alternative to the Spencer cohomological growth conditions imposed in [40,41]. Pseudo-groups having too large a fiber dimension r n (or, loosely, too many independent parameters), e.g., symplectomorphisms, [54], will, typically, act transitively on (a dense open subset of) J n , and thus possess no non-constant invariants.…”

Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

“…The invariant forms µ a J on G (n) satisfy the linear system (12) L (n) (Z i , µ a J ) = 0 obtained by replacing z i by Z i and ζ a J by µ a J in the linearized determining equations (11). In accordance with [23], we refer to (12) as the lifted determining equations for the pseudo-group. Theorem 4.2.…”

“…Theorem 4.2. The structure equations of the invariant coframe for a Lie pseudo-group G are obtained by restricting the diffeomorphism structure equations (8)(9) to the space of solutions of the lifted determining equations (12).…”

“…Unlike Lie groups, to this day, there is no generally accepted abstract object that represents an infinitedimensional pseudo-group, and so, like Lie and Cartan, [3], we can only study them in the context of their action on a manifold. This makes the subject considerably more difficult than the finite-dimensional case, and a significant effort has been made in establishing a proper rigorous foundation for pseudo-groups, [9,12,13,14,25,28].…”