Galois groups and connection matrices of -difference equations

Abstract

We study the Galois group of a matrix q-difference equation with rational coefficients which is regular at 0 and ∞, in the sense of (difference) Picard-Vessiot theory, and show that it coincides with the algebraic group generated by matrices C(u)C(w)−1 u,w ∈ C∗ , where C(z) is the Birkhoff connection matrix of the equation. 1. Differential algebra The notion of the Galois group of a linear ordinary differential equation or a holonomic system of such equations is well known. For example, consider the differential equation (1.1) df(z) dz = a(z)f(z), where a is an N × N -matrix valued function and f is an unknown C -valued function of one complex variable z (both are assumed holomorphic in z in a certain region). To define the Galois group, one fixes a field of functions F containing the coefficients of the equation and invariant under d/dz. Let L be the field generated over F by all solutions of the system. This field is invariant under d/dz. The Galois group G of the system is, by definition, the group of all automorphisms g of the field L such that g fixes all elements of F and [g, d/dz] = 0. The group G is naturally isomorphic to a linear algebraic group over C. Indeed, let f1, ..., fN ∈ L be a basis of the space of solutions of the system. Then for any g ∈ G, g(fi) are also solutions of the system, so g(fi) = ∑ j gijfj. Moreover, g is uniquely determined by the matrix gij . Thus we get an embedding of G into GLN(C), given by g → {gij} (superscript t means transposition). It can be shown that the image of this map is a closed subgroup. If we replace the basis f1, ..., fN by another basis, this embedding will be conjugated by the corresponding transformation matrix. The properties of group G are the subject of Picard-Vessiot (or differential Galois) theory, and are described in [Ko,R,Ka]. This theory allows to prove that certain linear differential equations of second and higher order cannot be reduced Received by the editors April 6, 1995. 1991 Mathematics Subject Classification. 12H10;39A10. c ©1995 American Mathematical Society