Tannakian Categories with Semigroup Actions

Ostrowski’s theorem implies that log(x), log(x + 1), . . . are algebraically independent over C(x). More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution y and particular transformations of y, such as derivatives of y with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form Nn × Z/n1Z × . . . × Z/nrZ on Tannakian categories. This is the class of semigroups that appear in many applications.