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

Abstract

Abstract. In this review article we discuss four recent methods for computing MaurerCartan structure equations of symmetry groups of differential equations. Examples include solution of the contact equivalence problem for linear hyperbolic equations and finding a contact transformation between the generalized Hunter-Saxton equation and the Euler-Poisson equation.

Differential invariant algebras of Lie pseudo-groups

“…Applications of these results and techniques can be found in [9,10,38,54]. The reader is advised to consult these papers before delving deeply into the detailed constructions and proofs presented here.…”

Pseudo-Groups, Moving Frames, and Differential Invariants

“…Examples of this procedure can be found in [13,58]; see also [48] for a comparison with other approaches. …”

“…Applications have included complete classifications of differential invariants and their syzygies, [56], equivalence and symmetry properties of submanifolds, rigidity theorems, invariant signatures in computer vision, [2,6,9,54], joint invariants and joint differential invariants, [7,54], rational and algebraic invariants of algebraic group actions, [26,27], invariant numerical algorithms, [30,55,72], classical invariant theory, [3,53], Poisson geometry and solitons, [41,42,43], and the calculus of variations, [31]. New applications of these methods to computation of symmetry groups and classification of partial differential equations can be found in [40,47,48]. Maple software implementing the moving frame algorithms, written by E. Hubert, can be found at [25] Our main goal in this contribution is to survey the extension of the moving frame theory to general Lie pseudo-groups recently put forth by the authors in [57,58,59,60], and in [13,14] in collaboration with J. Cheh.…”

“…The remarkable fact is that the Maurer-Cartan forms satisfy an "invariantized" version of the linear infinitesimal determining equations for the pseudo-group, and, as a result, we can immediately produce an explicit form of the pseudo-group structure equations. Application of these results to the design of a practical computational algorithm for directly determining the structure of symmetry (pseudo-)groups of partial differential equations can be found in [4,13,14,48]. Assuming freeness of the prolonged pseudo-group action at sufficiently high order, the explicit construction of the moving frame is founded on the Cartan normalization procedure associated with a choice of local cross-section to the pseudo-group orbits in J ∞ (M, p).…”

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

“…Moreover, isomorphic pseudo-group actions may admit non-isomorphic Cartan structure equations. We refer the reader to [60,76,91,92] for further details.…”