Tannakian Categories With Semigroup Actions

Abstract

Abstract. Ostrowski's theorem implies that log(x), log(x + ), . . . are algebraically independent over C(x). More generally, for a linear di erential or di erence equation, it is an important problem to nd all algebraic dependencies among a non-zero solution y and particular transformations of y, such as derivatives of y with respect to parameters, shi s 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 di erential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a nite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we nd a nite set of axioms that characterizes actions of semigroups that are nite free products of semigroups of the form N n ×Z n Z×⋅ ⋅ ⋅×Z n r Z on Tannakian categories. is is the class of semigroups that appear in many applications.

A topos-theoretic view of difference algebra

“…This context naturally leads to the notion of representations of affine difference algebraic groups as difference comodules of the associated difference Hopf algebra, allowing us to compare to the previous work of [51] and [15].…”