Tannakian Categories, Linear Differential Algebraic Groups, and Parametrized Linear Differential Equations

Tannaka’s theorem (cf. [19]) states that a linear algebraic group is determined by its category of representations. The problem of recognizing when a category is the category of representations of a linear algebraic group (or, more generally, an affine group scheme) is attacked via the theory of neutral Tannakian categories (see [18], [9]). This theory allows one to detect the underlying presence of a linear algebraic group in various settings. For example, the Galois theory of linear differential equations can be developed in this context (see [9], [10], [15]). In [2], Cassidy introduced the concept of a differential algebraic group and in [3] studied the representation theory of linear differential algebraic groups. Building on this work, we proved an analogue of Tannaka’s theorem for linear differential algebraic groups (see [17]). In the present paper we develop the notion of a neutral differential Tannakian category and show that this plays the same role for linear differential algebraic groups that the theory of neutral Tannakian categories plays for linear algebraic groups. As an application, we are able to give a categorical development of the theory of parametrized linear differential equations that was introduced in [4]. Another approach to the Galois theory of systems of linear differential equations with parameters is given in [1], where the authors study Galois groups for generic values of the parameters. Also, it is shown in [8] that over the field C(x) of rational