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Tannakian Categories With Semigroup Actions
- Source :
- Canadian Journal of Mathematics. 69:687-720
- Publication Year :
- 2017
- Publisher :
- Canadian Mathematical Society, 2017.
-
Abstract
- Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ 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. 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 $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.<br />Comment: minor revision
- Subjects :
- Class (set theory)
Pure mathematics
Semigroup
General Mathematics
010102 general mathematics
Braid group
Tannakian category
Group Theory (math.GR)
Mathematics - Commutative Algebra
Commutative Algebra (math.AC)
01 natural sciences
010101 applied mathematics
Linear differential equation
Mathematics::Category Theory
FOS: Mathematics
0101 mathematics
Mathematics - Group Theory
Finite set
Differential (mathematics)
Axiom
Mathematics
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 69
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....54a1133466d4832e59b3d5b0f24955ec