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

In this review article we discuss four recent methods for computing Maurer-Cartan 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.


Introduction
The theory of Lie groups has greatly influenced diverse branches of mathematics and physics. The main tool of the theory, Sophus Lie's infinitesimal method [30], establishes connection between continuous transformation groups and algebras of their infinitesimal generators. The method leads to many techniques of great significance in studying the group-invariant solutions and conservation laws of differential equations [42,21,36,4,28]. Application of Lie's infinitesimal method to concrete systems of differential equations requires analysis and integration of over-determined defining systems for symmetry algebras. Additional integrations need to be done when differential invariants and operators of invariant differentiations are computed via the infinitesimal method. Also, complexity of defining systems in symmetry analysis of classes of differential equations sometimes is high enough to make the full study of all branches of classification trees very hard.
An alternative approach for studying Lie (pseudo-)groups was developed byÉlie Cartan [6,7,8]. His theory is based on characterizing transformations from a pseudo-group by means of a set of invariant differential 1-forms called Maurer-Cartan forms. Expressions of exterior differentials of Maurer-Cartan forms in terms of the forms themselves yield the Cartan structure equations for the pseudo-group. These equations contain all information about the pseudo-group, in particular, they give all differential invariants and operators of invariant differentiations. The knowledge of Maurer-Cartan forms and differential invariants enables one to solve equivalence problems for classes of differential equations and to find mappings between equivalent equations. An important feature of the method is that it does not require integration, and allows one to find Maurer-Cartan forms by means of only differentiation and linear algebra operations.
Nowadays there are different methods for computing Maurer-Cartan forms and structure equations of symmetry pseudo-groups of differential equations. In this review paper we discuss four of these methods. We restrict our attention to symmetries of partial differential equations only. For applications of Cartan's method to symmetries of ordinary differential equations we refer, e.g., to [24,25,27,17,19,3,37,11,12].
2 Apriori known geometric properties of a given differential equation The first approach is presented in [45,26,27,1,2,3,10,13]. It is based on apriori known information about geometric properties of a given differential equation. We illustrate the method on the example of Liouville's equation It is a hyperbolic equation of second order in two independent variables. From the geometric theory of such equations (see, e.g., [3]) it follows that there exists a coframe (a collection of independent 1-forms) ϑ, ω i , i ∈ {1, . . . , 4}, on R 5 with coordinates (x, y, u, p, q), such that a local section σ : (x, y) → (x, y, u(x, y), p(x, y), q(x, y)) of the trivial bundle R 5 → R 2 , (x, y, u, p, q) → (x, y), provides a local solution u(x, y) of equation (1) whenever σ * ϑ = 0, σ * (ω 1 ∧ ω 2 ) = 0, and σ * (ω 3 ∧ ω 4 ) = 0. Indeed, we take Then a local diffeomorphism Φ : R 5 → R 5 , Φ : (x, y, u, p, q) → (x, y, u, p, q), is a contact symmetry of equation (1) whenever it preserves the ideal generated by the contact form ϑ and the ideals generated by the pairs ω 1 , ω 2 and ω 3 , ω 4 modulo the contact ideal. That is, Φ must satisfy the following condition: . This is a standard set-up for Cartan's equivalence method [6,7,8,15,27,37]. Procedures of this method give the Maurer-Cartan forms , z 1 and z 2 are arbitrary parameters, with the structure equations Thus the method is quite simple. It does not require writing out defining systems for symmetry generators. Also, it gives Maurer-Cartan forms, differential invariants, and invariant derivatives for symmetry groups explicitly. But a lot of preliminary work needs to be done before the method becomes applicable to a given differential equation. It is neccessary to find a formulation of the equation in terms of ideals of exterior forms convenient for setting up the equivalence problem for coframes.

Taylor series expansion of defining equations for infinitesimal generators of transitive Lie pseudo-groups
The second method was provided in [31,32]. It extracts information about Cartan's structure equations from defining systems for infinitesimal generators. The basic point is that in the case of a transitive symmetry group the explicit form of solutions of the defining system is not required for computing coefficients of the structure equations. These coefficients depend on the finite order Taylor series expansions of the generators, while the information about the Taylor series can be obtained from the involutive form of the defining system. When the defining system for the infinitesimal generator X = n i=1 ξ i ∂ ∂x i of a transitive Lie pseudo-group G is of the first order, the procedure of the method is as follows. Let P 1 be the set of all parametric derivatives of the defining system. Denote for all ∂ξ i ∂x j ∈ P 1 . Then the involutive form of the defining system is for all principal derivaties. As it is shown in [32], in this case Cartan's structure equations of the pseudo-group G have the form , where x 0 is any non-singular point of the defining system (4). [31,Example 6].) Consider the Lie pseudo-group on R 2 whose infinitesimal generator X = ξ ∂ x + η ∂ y satisfies the defining system ξ x = 1 y η, ξ y = 0, η y = 1 y η.

Example 1. (See
In this system, the only parametric derivative of the first order is η x . We denote it by φ 1 and add the only parametric equation (3): Therefore, in any non-singular point (x 0 , y 0 ) with y 0 = 0 we have A 2 This yields the structure equations Any value y 0 = 0 is suitable as an initial data point, so we choose y 0 = 1.
Additional work should be done when the order of the defining system is greater then 1 [32,Section 4.4].  (1) the coefficients of the infinitesimal generator X = ξ ∂ x + η ∂ y + τ ∂ u satisfy the defining system in involutive form The algorithm of [32] gives the following form of the Cartan structure equations: This result coincides with the previous one: the substitution of (2) gives (6).
The method has the following properties. It is not universal since it is not applicable to differential equations with intransitive symmetry pseudo-groups. Also, it does not give Maurer-Cartan forms and differential invariants explicitly. An integration should be used to find them from the structure equations [14, Section 7.6].

Invariantized defining equations for Maurer-Cartan forms
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. Let . . , 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 is the inverse matrix for the Jacobian matrix (Z α j ). The structure equations for D (∞) (M ) have the form the Jacobian matrix of the vector µ H of power series in the variables H = (H j ). Let V = ζ α ∂ z α be the infinitesimal generators of a sub-pseudo-group G ⊂ D(M ). They are characterized by the defining equations If G is the symmetry group of a system of differential equations, then (9) are (the involutive completion of) the usual determining equations obtained through Lie's infinitesimal technique. The method is based on the following theorems [38]: obtained by replacing z i by Z i and ζ α I by µ α I in the determining equations (9).
Theorem 2. The structure equations of the invariant coframe for a Lie pseudo-group G are obtained by restricting the diffeomorphism structure equations (7), (8) to the space of solutions of the equations (10).
These results allow to find the structure equations of the symmetry group of a system of differential equations from an involutive form of infinitesimal defining equations. The required computations rely exclusively on linear algebra and differentiation, and can be readily implemented in any standard symbolic computation package.
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. For differential equations with infinite symmetry pseudo-groups the method produces infinite sets of Maurer-Cartan forms and infinite systems of structure equations. Also, the Maurer-Cartan forms obtained by this method depend on both source and target variables of the diffeomorphism pseudo-group. Therefore, additional work needs to be done to obtain finite sets of Maurer-Cartan forms and to express them in terms which are suitable for further implementations such as finding transformations between equivalent equations.

The moving coframe method
Finally, the fourth approach is based on the moving coframe method provided in [12]. Applied to contact symmetries of differential equations of the second order, the method has the following outline. Consider the bundle J 2 (E) of the second order jets of local sections of the trivial bundle E = R n × R → R n . Contact transformations ∆ : . Therefore, Maurer-Cartan forms for the pseudo-group Cont(J 2 (E)) of contact transformations on J 2 (E) can be easily found using Cartan's equivalence method, see, e.g., [35]. These forms are is the inverse matrix for the matrix (b i j ). The structure equations of Cont(J 2 (E)) have the following form: A differential equation R of the second order is a subbundle of J 2 (E). Let ι : R → J 2 (E) be the inclusion map. Then we can find the Maurer-Cartan forms for the pseudo-group Sym(R) of contact symmetries of R from the restrictions θ 0 = ι * Θ 0 , θ i = ι * Θ i , ξ i = ι * Ξ i , σ ij = ι * Σ ij by standard procedures of Cartan's equivalence method, see [12,33,34] for details.
The moving coframe method enables one to find Maurer-Cartan forms, differential invariants, and operators of invariant differentiations for symmetry pseudo-groups explicitly, in contrast to the method of [32]. It does not require integration at all, and, unlike the methods of [32,38,39,9], does not use infinitesimal defining systems. Also, the method is universal since it is applicable to any differential equation. The price of these advantages is lengthy and intricate computations. Nevertheless the method allows one to solve effectively problems concerned with symmetry classification and equivalence of differential equation. We illustrate it in the two following sections.

Contact equivalence problem for linear hyperbolic equations
In [35] the moving coframe method is used to solve the local equivalence problem for the class of linear second order hyperbolic equations in two independent variables w.r.t. the pseudo-group of contact transformations. This class has been studied for more than two centuries. In [29] P.S. Laplace found semiinvariants H = −T t + T X + U and K = −X x + T X + U . These functions are invariants of the sub-pseudo-group u → λ(t, x)u, λ = 0. Laplace proved that equation (13) (13) with non-trivial finite-dimensional subgroups of symmetry pseudo-groups [41], [42,Section 9.2].
The solution of the contact equivalence problem for the class (13) is found in [35]: C 5 consists of all equations (13) such that P ≡ const, Q t ≡ 0, and Q x = 0; C 6 consists of all equations (13) such that P ≡ const and Q ≡ const.
Every equation from the subclass C 1 is equivalent to the linear wave equation u tx = 0. Every equation from the subclass C 6 is equivalent to the equation when Q = 0, or to the Euler-Poisson equation when Q = 0.
For the subclass C 2 , the basic invariants are P , Q, and J = (H t P t − HP tt ) H −1 P −2 t , the operators of invariant differentiation are D 1 = P −1 t D t and D 2 = P t H −1 D x . For C 3 , the basic invariants are P , Q, and L = ( x D x . Two equations from the subclasses C 2 -C 5 are locally equivalent to each other if and only if they have the same functional dependencies among the basic invariants and their invariant derivatives up to the second order. Different results were stated in [23] and [22]: Theorem 4 ( [23]). The functions P , Q, are a basis of the complete set of invariants of (13). Any other differential invariant is a function of these basic invariants and their invariant derivatives.

Theorem 5 ([22]). A basis of invariants for equations (13) consists of the invariants
or of invariants of the alternative basis The operators of invariant differentiations are taken in [22] in the following form: D 1 = P −1 t D t and D 2 = P −1 x D x . The functions of Theorems 4, 5 do not provide bases of invariants for the whole class of equations (13). To prove this, we consider the following equation: with p(t) = 0, q(t) = 0, p ′ (t) = 0, and q ′ (t) = 0. For this equation we have P = p(t), Q = q(t), .
Since J depends on x explicitly, it is functionally independent of P , Q, J 1 3 , J 2 3 , J 3 3 , I, and Q. Therefore, the functions from Theorem 4 and the first set of functions from Theorem 5 do not provide a basis of differential invariants for the whole class (13).
Otherwise, for Moutard's equation we have Q = Q, P = 1, so P t = P x = I = 0. Therefore the invariant J and the operators x D x are not defined for this equation. This proves that the both sets of functions from Theorem 5 do not provide bases of invariants for the subclass (16).

Remark 1.
As it is shown in [35], the symmetry pseudo-groups of equations (13) from the subclasses C 2 -C 5 are intransitive. Therefore, the moving coframe method is shown to be applicable to study intransitive symmetry pseudo-groups as well.

Linearizability of the generalized Hunter-Saxton equation
In this section, we use the moving coframe method to study the generalized Hunter-Saxton equation This equation has a number of applications in the nonlinear instability theory of a director field of a liquid crystal [20], in geometry of Einstein-Weil spaces [46,18], in constructing partially invariant solutions for the Euler equations of an ideal fluid [16], and has been a subject of many recent investigations. In the case κ = 1 2 the general solution [20], the tri-Hamiltonian formulation [40], the pseudo-spherical formulation and the quadratic pseudo-potentials [44] have been found. The conjecture of linearizability of equation (17) in the case κ = −1 has been made in [16]. In [43] a formula for the general solution of (17) has been proposed. This formula uses a nonlocal change of variables.
We prove that equation (17) is equivalent under a contact transformation to the Euler-Poisson equation (14) with P = 2 (1 − κ) and Q = 2 κ: Also, we find the general solution of (17) in terms of local variables. Using the moving coframe method, we obtain the structure equations for the symmetry pseudo-group of equation (18) in the form is the right-hand side of equation (18), while a and b are arbitrary non-zero constants. The forms σ 11 , . . . , π 2 are too long to be written out in full here. We write equation (17) and its Maurer-Cartan forms in tilded variables, then similar computations give where R 1 is the right-hand side of equation (17) written in the tilded vatiables, while a and b are arbitrary non-zero constants. The forms σ 11 , . . . , π 2 are too long to be written out in full. The structure equations for (17) differ from (19) only in replacing θ 0 , . . . , π 2 by their tilded counterparts. Therefore, results of Cartan's method (see, e.g., [37,Theorem 15.12]) yield the contact equivalence of equations (17) and (18). Since the Maurer-Cartan forms for both symmetry groups are known, the equivalence transformation Ψ : (t, x, u, u t , u x ) → ( t, x, u, u t , u x ) can be found from the requirements Ψ * θ 0 = θ 0 , Ψ * θ 1 = θ 1 , Ψ * θ 2 = θ 2 , Ψ * ξ 1 = ξ 1 , and Ψ * ξ 2 = ξ 2 : takes the Euler-Poisson equation (18) to the generalized Hunter-Saxton equation (17) (written in the tilded variables).
Equation (18) has an intermediate integral, and its general solution can be found in quadratures. Indeed, for the invariants of this equation we have P + Q = 2, therefore the Laplace t-transformation, [42, Section 9.3], takes equation (18) to a factorizable linear hyperbolic equation. Namely, we consider the system Substituting (20) into (21) yields equation (18), while expressing u from (21) and substituting it into (20) gives the equation with the trivial Laplace semi-invariant H. Hence, the substitution takes equation (22) into the equation w t = −2 (κ − 1) (κ (t + x)) −1 w.
Integrating (24) and (23), we have the general solution for equation (22): where S(t) and R(x) are arbitrary smooth functions of their arguments. Then equation (21) gives the general solution for equation (18): This formula together with the contact transformation from Theorem 6 gives the general solution for the generalized Hunter-Saxton equation (17) in a parametric form: Hence, we obtain the general solution of equation (17) without employing nonlocal transformations.

Discussion
Cartan's method of equivalence in its different incarnations is a powerful tool in the study of symmetry groups of differential equations. Each of the four approaches discussed has its own advantages and shortcomings. While the first and the second approaches are not universal, the third and the fourth approaches provide an effective technique which is valid irrespective of geometric properties of a given differential equation or of intransitivity of the symmetry pseudogroup. The last approach, the moving coframe method, is applicable to solving the equivalence problems, finding differential invariants and operators of invariant differentiation for classes of differential equations where the power of the infinitesimal method is restricted by complexity of analysis of the defining systems for symmetry generators. Unlike the second and the third methods, the moving coframe method does not require analysis of the defining equations and enables one to find the Maurer-Cartan forms for symmetry pseudo-groups explicitly. The price of its power is lengthy and massive computations. It would be worthwhile to improve the algorithms of the basic steps involved in the method. We conclude the paper by mentioning evident prospects for the use of Cartan's method in studying structure of symmetry pseudo-groups of differential equations. The directions of future research would include, among others, simplification of solving symmetry classification problems and clarification of the group foliation technique with applications to construction of differentially-invariant solutions. Therefore, the further development and improvement of implementations of Cartan's method would be of great interest and significance for the study of geometry of differential equations.