Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

Abstract

ABSTRACT. A new method of constructing structure equations of Lie symmetry pseudo-groups of differential equations, dispensing with explicit solutions of the (infinitesimal) determining systems of the pseudo-groups, is presented, and illustrated by the examples of the Kadomtsev-Petviashvili and Korteweg-de-Vries equations.

Structure of Symmetry Groups via Cartan's Method: Survey of Four Approaches

“…. , i m ) (our notation for multi-indexes differs slightly from those of [38,39,9]). Then, as it is shown in [38], the Maurer-Cartan forms for…”

“…Unlike the previous two methods, the method of [38,39,9] is universal since it is applicable to any differential equation. It requires analysis of the defining systems for infinitesimal generators and its reduction to the involutive form.…”

“…The third method is developed in [38,39,9]. It is based on use of invariantized defining equations for Maurer-Cartan forms of Lie pseudo-groups.…”

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

“…In the case of finite-dimensional Lie group actions, the reformulation of a moving frame, [14,30], as an equivariant map back to the Lie group, [28], proved to be amazingly powerful, sparking a host of new tools, new results, and new applications, including complete classifications of differential invariants and their syzygies, [31,69,71], equivalence, symmetry, and rigidity properties of submanifolds, [28], computation of symmetry groups and classification of partial differential equations, [49,59], invariant signatures in computer vision, [4,8,12,67], joint invariants and joint differential invariants [9,67], rational and algebraic invariants of algebraic group actions [32,33], invariant numerical algorithms [38,68,95], classical invariant theory [5,66], Poisson geometry and solitons [50,51,52], the calculus of variations and geometric flows, [39,70], invariants and covariants of Killing tensors, with applications to general relativity, separation of variables, and Hamiltonian systems, [56,57], and invariants of Lie algebras with applications in quantum mechanics, [10]. Subsequently, building on the examples presented in [27], a comparable moving frame theory for general Lie pseudo-group actions was established, [72,73,74], and applied to several significant examples, [19,20].…”

“…Moreover, the structure equations are found by restricting the explicit diffeomorphism structure equations to the kernel of a linear algebraic system directly related to the linearized determining equations for the pseudo-group's infinitesimal generators. A large number of examples arise as symmetry groups of differential equations, and we provide a quick review of the classical Lie infinitesimal method of calculating symmetry groups, [64], and then show how, using the preceding result, the structure of the symmetry group of a system of differential equations can be directly found without integration of the determining equations, [19,72]. We then review the extension of the equivariant moving frame theory to pseudo-group actions on jets of submanifolds, leading to an algorithmic procedure for constructing the differential invariants and invariant differential forms, [20,73].…”