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# Tannakian Categories with Semigroup Actions

Published:2016-09-16
Printed: Jun 2017
• Alexey Ovchinnikov,
CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Queens, NY 11367, USA
• Michael Wibmer,
University of Pennsylvania, Department of Mathematics, 209 South 33rd Street, Philadelphia, PA 19104, USA
 Format: LaTeX MathJax PDF

## Abstract

Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ 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, 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 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 $\mathbb{N}^n\times \mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.
 Keywords: semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters
 MSC Classifications: 18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 12H10 - Difference algebra [See also 39Axx] 20G05 - Representation theory 33C05 - Classical hypergeometric functions, ${}_2F_1$ 33C80 - Connections with groups and algebras, and related topics 34K06 - Linear functional-differential equations

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