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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
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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 semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters
MSC Classifications: 18D10, 12H10, 20G05, 33C05, 33C80, 34K06 show english descriptions Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
Difference algebra [See also 39Axx]
Representation theory
Classical hypergeometric functions, ${}_2F_1$
Connections with groups and algebras, and related topics
Linear functional-differential equations
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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