Abstract view
Tannakian Categories with Semigroup Actions


Published:20160916
Printed: Jun 2017
Alexey Ovchinnikov,
CUNY Queens College, Department of Mathematics, 6530 Kissena Blvd, Queens, NY 11367, USA
Michael Wibmer,
University of Pennsylvania, Department of Mathematics, 209 South 33rd Street, Philadelphia, PA 19104, USA
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 nonzero 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.
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 functionaldifferential 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 functionaldifferential equations
