1. Tannakian Categories With Semigroup Actions
- Author
-
Michael Wibmer and Alexey Ovchinnikov
- Subjects
Class (set theory) ,Pure mathematics ,Semigroup ,General Mathematics ,010102 general mathematics ,Braid group ,Tannakian category ,Group Theory (math.GR) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,010101 applied mathematics ,Linear differential equation ,Mathematics::Category Theory ,FOS: Mathematics ,0101 mathematics ,Mathematics - Group Theory ,Finite set ,Differential (mathematics) ,Axiom ,Mathematics - Abstract
Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. 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. 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 $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications., Comment: minor revision
- Published
- 2017