Tannakian Categories With SemigroupActions

Ovchinnikov, Alexey ; Wibmer, Michael

Canadian journal of mathematics, Tome 69 (2017) no. 3, pp. 687-720

Résumé

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.

Détail

DOI : 10.4153/CJM-2016-011-0

Mots-clés : 18D10, 12H10, 20G05, 33C05, 33C80, 34K06, semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters