Tannakian Categories With Semigroup Actions

Abstract

AbstractA theorem of Ostrowski 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 this 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 ûnite collection of axioms that characterizes such actions to apply them to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are ûnite free products of semigroups of the form on Tannakian categories. This is the class of semigroups that appear in many applications.