Tannakian Categories With Semigroup Actions. (1st June 2017)

• Record Type: Journal Article
• Authors: Ovchinnikov, Alexey; Wibmer, Michael
• Is Part Of: Canadian journal of mathematics. Volume 69:Number 3(2017)
• Page Start: 687
• Page End: 720
• Publication Date: 2017-06-01
• Subjects: 18D10 -- 12H10 -- 20G05 -- 33C05 -- 33C80 -- 34K06; semigroup actions on categories -- Tannakian categories -- difference algebraic groups -- differential and difference equations with parameters; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2016-011-0
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: A theorem of Ostrowski implies that $\log \left( x \right), \, \log \left( x+1 \right), \, .\, .\, .$ are algebraically independent over $\mathbb{C}\left( x \right)$ . 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 finite 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 finite free products of semigroups of the form ${{\mathbb{N}}^{n}}\, \times \, \mathbb{Z}/{{n}_{1}}\mathbb{Z}\, \times \, .\, .\, .\, \mathbb{Z}/{{n}_{r}}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.