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Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems

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Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Symbolic Computation (1998) 26, 355–379 Article No. sy980218 Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems IAN G. LISLE† AND GREGORY J. REID‡ †School of Mathematics and Statistics, University of Canberra, Australia ‡Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby BC, Canada An algorithm is described which uses a finite number of differentiations and linear op- erations to determine the Cartan structure of a transitive Lie pseudogroup from its infinitesimal defining equations. In addition, an algorithm is presented for determining from the infinitesimal defining system whether a Lie pseudogroup has essential invariants. If such invariants exist, the pseudogroup is intransitive. These methods make feasible the calculation of the Cartan structure of infinite Lie pseudogroups of symmetries of differential equations. The structure of the symmetry pseudogroup of the KP equation is presented. c 1998 Academic Press 1. Introduction This work is concerned with the algorithmic determination of geometrical and structural properties of Lie pseudogroups, that is, properties which are invariant under coordinate changes and Lie pseudogroup isomorphism respectively. The collection of volume pre- serving local diffeomorphisms on Rn is an example of a pseudogroup, and satisfies most but not all the axioms of a group. In particular, composition is not defined for all pairs of local diffeomorphisms, since the range of one diffeomorphism may not overlap the do- main of a second. Since a member X = f(x) of the collection preserves a volume element dx1 dxn, it must satisfy the nonlinear analytic partial differential equation (PDE) ∧···∧ 1 1 Xx1 Xxn . ··· . =1 (1.1) n n X 1 X n x x ··· i ∂Xi where Xxj = ∂xj . A pseudogroup whose local diffeomorphisms are analytic and satisfy an analytic system of PDEs will be called a Lie pseudogroup, and the system of PDEs will be called the pseudogroup defining system. To obtain the infinitesimal defining system of this Lie pseudogroup, the pseudogroup defining system is linearized about its identity transformation. For example, by substi- tuting Xi = xi + εξi(x)+O(ε2) into (1.1), the infinitesimal defining system 1 2 n ξ 1 + ξ 2 + + ξ n =0 (1.2) x x ··· x 0747–7171/98/090355 + 25 $30.00/0 c 1998 Academic Press 356 I.G. Lisle and G.J. Reid is obtained. It is a linear homogeneous PDE for variables ξi, components of a vector field 1 2 n ξ ∂x1 + ξ ∂x2 + + ξ ∂xn , whose flow diffeomorphisms are in the pseudogroup. When n = 1, equations··· (1.1) and (1.2) can each be integrated, giving X = x + a and ξ = b respectively, where a and b are constants. Thus n = 1 corresponds to the finite- (one-) parameter Lie pseudogroup of local translations. However, any case with n>1isan infinite Lie pseudogroup, that is, a pseudogroup with no finite parametrization. In this case the pseudogroup defining system cannot be explicitly integrated. Thus a systematic analysis of Lie pseudogroups should be based on the defining systems rather than on knowledge of the solutions of the defining systems. We address the problem of systematically determining structural properties of Lie pseudogroups—that is, properties preserved under pseudogroup isomorphism (see Sec- tion 2). We present and justify a constructive algorithm for determining the Cartan structure of a transitive Lie pseudogroup from its linear infinitesimal defining system. By an algorithm being constructive we mean that it attains its goal in a finite number of steps, each of which only involves a differentiation or a linear operation. In particular a constructive algorithm does not involve integration, which is in general a heuristic pro- cess. As part of our method, we constructively determine certain geometric features of a Lie pseudogroup, including the dimension of its orbits and its linear isotropy group. The linear systems we deal with generally involve functions as their coefficients. Unless otherwise stated we assume that these functions belong to some computable domain (e.g. the rational functions over Q, or some computable extension of this field). In this way we will always be able to determine whether a given coefficient is zero or nonzero and thus effectively perform Gaussian elimination on such systems. In addition we make the following: Blanket hypothesis.The terms ‘diffeomorphism’, ‘differential equation’ and ‘vector field’ are henceforth understood to be real analytic without exception. We shall use the summation convention throughout, and also use the convention that bold face roman letters X, Y etc. denote vector fields. We were motivated to develop the approach of this paper by our success (Reid et al., 1992) in developing a constructive algorithm to determine the structure of finite parameter Lie pseudogroups from their infinitesimal defining systems. This algorithm is particularly well suited to computer algebra implementation since it bypasses the heuristic step of integrating infinitesimal defining equations used by other methods. Reid et al. (1992) generalized this method to finding commutation relations of infinite Lie pseudogroups, but it appears difficult to extract structural information from this gener- alization. For finite Lie pseudogroups, a great simplification is the linear ‘infinitesimal’ theory. i By seeking vector fields X = ξ (x)∂xi whose associated one-parameter groups of local diffeomorphisms are in the pseudogroup, one obtains a linear homogeneous defining sys- tem for the components ξi (the ‘infinitesimals’). The resulting vector fields X constitute a Lie algebra. In the finite-dimensional case the Lie algebra can be resolved with respect to a basis Xk, so that closure under commutator bracket takes the form k [Xi, Xj]=cijXk, (1.3) k where the constants cij are the structure constants of the Lie algebra. The great contribu- Geometry and Structure of Lie Pseudogroups 357 k tion of Lie was to prove that cij encode all the local structure of a finite Lie pseudogroup. Calculation of structure is then a linear problem. For symmetry analysis of differential equations (Bluman and Kumei, 1989; Olver, 1993; Ovsiannikov, 1982) the infinitesimal approach is essential. For almost all PDE systems of interest the infinitesimal defining system of the Lie symmetry pseudogroup can be algorithmically derived (Clarkson and Mansfield, 1994). Many symbolic programs are now available for deriving and attempting to integrate these infinitesimal defining systems (Hereman, 1994). Recently methods have become available which extract information directly from the infinitesimal defining systems without integration. Reid (1990, 1991b) and Reid et al. (1992) and independently Schwarz (1992a, 1992b), showed how to obtain the dimension of the symmetry algebra from the infinitesimal defining system, while Reid k (1990, 1991b) and Reid et al. (1992) showed how to obtain the structure constants cij in the finite-parameter case. These authors also provided computer algebra implementations of these methods (Reid, 1991b; Schwarz, 1992a). The central idea of such integration-free methods is that the infinitesimal defining system can, by a process of Gauss reduction and appending integrability conditions, be constructively reduced to a (coordinate dependent) canonical form (Carr`a-Ferro, 1987; Mansfield, 1991; Reid, 1991b; Schwarz, 1992a) for which a local existence and uniqueness theorem is available (Janet, 1920; Riquier, 1910). The coordinate independent counterpart of these approaches involves the reduction of systems of PDEs to involutive form (Bryant et al., 1991; Pommaret, 1978). The canonical form and involutive form of a system of PDEs are related in that it has been shown that the canonical form of a system of linear PDEs can be constructively prolonged to yield an involutive system (Carr`a-Ferro and Duzhin, 1993; Mansfield, 1991; Mansfield, 1996). The reduction algorithms above will be of crucial use in enabling us to extract the Cartan structure of infinite Lie pseudogroups. Lie did not develop a structure theory for infinite Lie pseudogroups, although he be- lieved (Lie, 1891) that infinitesimal methods were still appropriate. Cartan (1905, 1908, 1937a) and Vessiot (1904) devised structure theories of Lie pseudogroups. Like Lie they defined their pseudogroups as the general solution of analytic defining equations. Car- tan’s structure theory (Kamran, 1989) is stated in terms of one-forms ωi invariant under the action of the pseudogroup: taking exterior derivatives, Cartan obtains dωk = ak πρ ωi 1 ck ωi ωj (1.4) iρ ∧ − 2 ij ∧ ρ k k where π are certain additional one-forms. The structure coefficients are cij, aiρ.Ifthe k k pseudogroup is of finite type then the aiρ are absent, the cij are constant, and the Car- tan structure equations reduce to the Maurer–Cartan equations, which are dual to Lie’s commutation relations (1.3). The Cartan structure coefficients are, therefore, a direct gen- k k eralization of Lie’s cij. In the infinite case, aiρ terms appear. However, the unexpected difficulty that arises in the infinite case is the possible presence of essential scalar invari- ants. For instance in the pseudogroup X = x, Y = y+f(x), the invariant x is essential: it must appear in any realization of this pseudogroup. There is thus a fundamental division between transitive pseudogroups (no essential invariants) and intransitive pseudogroups (which possess essential invariants). The intransitive case is further complicated by the k k fact that the structure coefficients aiρ, cij can be functions of the essential invariants, so that the structure varies from point to point. In the transitive (including the finite parameter) case, they are structure constants. Cartan gave an algorithmic procedure for computing the one-forms ωi and calculating 358 I.G. Lisle and G.J. Reid the structure equations directly from the involutive form of the defining system of the pseudogroup. Although Cartan obtained many important structural results in this way, the method is difficult to apply to symmetry analysis of differential equations.
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