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Tannakian Categories With SemigroupActions

Published online by Cambridge University Press:  20 November 2018

Alexey Ovchinnikov  and
Michael Wibmer
Alexey Ovchinnikov
Affiliation:
CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Queens, NY 11367, USAand , CUNY Graduate Center, Department of Mathematics,365 Fifth Avenue, New York, NY 10016, USA e-mail: aovchinnikov@qc.cuny.edu
Michael Wibmer
Affiliation:
RWTH Aachen, 52056, Aachen, Germany and , University of Pennsylvania, Department of Mathematics, 209 South 33rd Street, Philadelphia, PA 19104, USA e-mail: wibmer@math.upenn.edu
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Abstract

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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.

Keywords

18D1012H1020G0533C0533C8034K06semigroup actions on categoriesTannakian categoriesdifference algebraic groupsdifferential and difference equations with parameters
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 3 , 01 June 2017 , pp. 687 - 720
Copyright
Copyright © Canadian Mathematical Society 2017

References

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