Galois theory of q-difference equations

Second order linear q-difference equations: Nonoscillation and asymptotics

“…in [1], [2], [3], [4], [6], [8], [10], [12], [16], [19], [24], [29]. For related topics see [11], [15], [18], [30] and the references therein. Note that (1.1) may be viewed as a special case of the linear dynamic equation…”

Commutation Relations and Discrete Garnier Systems

“…This formal construction may be replaced by a fundamental system of meromorphic solutions of either (1.1) or (1.6) specified by Theorem 2.11. For q-difference equations, in general (see [51]) the entries of any solution are elements of the field M(C)(l q , (e q,c ) c∈C * ).…”

$q$-Conjugacy classes in loop groups

“…We shall discuss the connection with holomorphic principal bundles over elliptic curves. In [11] it is observed that for q ∈ C × , |q| < 1, there is a functor from the category of split D q -modules (split means V ≃ grV ) over C{t}, the field of convergent Laurent series, to the category of vector bundles on E q (C) ≃ C × /q Z , which is bijective on isomorphism classes of objects and respects tensor product. Moreover it is also proved that split D q -modules over C{t} have the same classification as D q -modules over C((t)).…”

“…Proof. The argument in [11] still works, once we prove that any one-dimensional D q -module overk((t)) 0 is isomorphic to W 1 (r, λ) for some r ∈ Z and λ ∈ k × . This amounts to the fact that the multiplicative group U := {1 + a 1 t + a 2 t 2 + · · · |a i ∈k} ∩k((t)) 0 enjoys the property that any u ∈ U can be written as v −1 σ q v for some v ∈ U.…”