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Tannakian Categories With SemigroupActions

Published online by Cambridge University Press:  20 November 2018

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
Corresponding

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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

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References

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