Maurer–Cartan forms and the structure of Lie pseudo-groups

Abstract

Sur la théorie, si importante sans doute, mais pour nous si obscure, des ≪groupes de Lie infinis≫, nous ne savons rien que ce qui se trouve dans les mémoires de Cartan, premiere explorationà travers une jungle presque impénétrable; mais cell-ci menace de se refermer sur les sentiers déjà tracés, si l'on ne procède bientôtà un indispensable travail de défrichement.-André Weil, [76] Abstract.This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer-Cartan forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation.The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudo-group on submanifolds. The third paper [61] will apply Gröbner basis methods to prove a fundamental theorem on the freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.

Algebraic and Differential Invariants

“…A natural way to further advance the topic has been initiated in [52,50,51]. A methodological theory for pseudo-groups is introduced there in the same way as [14] started the topic for finite dimensional Lie groups.…”

“…Invariants of these group actions typically arise to reduce a problem or to decide if two objects, geometric or abstract, are obtained from one another by the action of a group element. [8,9,10,11,13,15,17,39,40,42,43,45,46,52,59] are a few recent references of applications. Both algebraic and differential invariant theories have become in recent years the subject of computational mathematics [13,14,17,40,60].…”

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].…”

“…The Maurer-Cartan forms of a Lie pseudo-group are explicitly constructed as invariant differential forms on the infinite pseudo-group jet bundle, [72]. 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.…”

“…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].…”