Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

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 il- lustrated by the examples of the Kadomtsev-Petviashvili and Korteweg-de-Vries equations. The theory of continuous groups of transformations created by Sophus Lie in the late nineteenth century has evolved to become one of the most important tools for geometric and algebraic study of general nonlinear partial differential equa- tions. Lie himself made no essential distinction between finite-dimensional Lie group actions and infinite-dimensional pseudo-group actions. However, since his time, the two subjects have developed in very different directions. The the- oretical foundations of finite-dimensional Lie groups and Lie algebras were well- established in the early twentieth century. In contrast, despite its evident impor- tance in both mathematics and applications, the basic theory for infinite-dimen- sional Lie pseudo-groups remains inrelatively primitive shape. Unlike Lie groups, to this day, there is no generally acceptedabstract object that represents an infinite- dimensional 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 con- siderably 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). Lie pseudo-groups appear in gauge theories, Hamiltonian mechanics, symplec- tic and Poisson geometry, conformal geometry of surfaces, conformal field the- ory, and geometry of real hypersurfaces, as symmetry groups of both linear and nonlinear partial differential equations arising in fluid mechanics, solitons, rela- tivity, etc., and as foliation-preserving groups of transformations. In general, a Lie pseudo-group G is defined in terms of a system R of (typically nonlinear) differential equations, called its determining system, whose solutions are the lo- cal diffeomorphisms constituting the pseudo-group. One immediate issue is to determine their local structure, which is usually expressed in the form of Maurer- Cartan structure equations, as in the case of finite-dimensional Lie groups. Both Lie's attempt to use his infinitesimal method based on the infinitesimal determin- ing system obtained by linearizing the determining system, and Cartan's method using intricate recursive prolongation of exterior differential systems are either limited in scope or impractical from the standpoint of applications. Along this