1,509 research outputs found
On globally nilpotent differential equations
In a previous work of the authors, a middle convolution operation on the
category of Fuchsian differential systems was introduced. In this note we show
that the middle convolution of Fuchsian systems preserves the property of
global nilpotence. This leads to a globally nilpotent Fuchsian system of rank
two which does not belong to the known classes of globally nilpotent rank two
systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven
whose differential Galois group is isomorphic to the exceptional simple
algebraic group of type $G_2.
Galois theory of q-difference equations
Choose with 0<|q|<1. The main theme of this paper is the
study of linear q-difference equations over the field K of germs of meromorphic
functions at 0. It turns out that a difference module M over K induces in a
functorial way a vector bundle v(M) on the Tate curve . As a corollary one rediscovers Atiyah's classification of
the indecomposable vector bundles on the complex Tate curve. Linear
q-difference equations are also studied in positive characteristic in order to
derive Atiyah's results for elliptic curves for which the j-invariant is not
algebraic over . A universal difference ring and a universal
formal difference Galois group are introduced. Part of the difference Galois
group has an interpretation as `Stokes matrices', the above moduli space is the
algebraic tool to compute it. It is possible to provide the vector bundle v(M)
on E_q, corresponding to a difference module M over K, with a connection
. If M is regular singular, then is essentially determined
by the absense of singularities and `unit circle monodromy'. More precisely,
the monodromy of the connection coincides with the action of
two topological generators of the universal regular singular difference Galois
group. For irregular difference modules, will have singularities and
there are various Tannakian choices for . Explicit
computations are difficult, especially for the case of non integer slopes.Comment: Corrected versio
Galois theory for iterative connections and nonreduced Galois groups
This article presents a theory of modules with iterative connection. This
theory is a generalisation of the theory of modules with connection in
characteristic zero to modules over rings of arbitrary characteristic. We show
that these modules with iterative connection (and also the modules with
integrable iterative connection) form a Tannakian category, assuming some nice
properties for the underlying ring, and we show how this generalises to modules
over schemes. We also relate these notions to stratifications on modules, as
introduced by A. Grothendieck in order to extend integrable (ordinary)
connections to finite characteristic. Over smooth rings, we obtain an
equivalence of stratifications and integrable iterative connections.
Furthermore, over a regular ring in positive characteristic, we show that the
category of modules with integrable iterative connection is also equivalent to
the category of flat bundles as defined by D. Gieseker. In the second part of
this article, we set up a Picard-Vessiot theory for fields of solutions. For
such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes
into account even nonreduced closed subgroup schemes of the Galois group scheme
on one hand and inseparable intermediate extensions of the Picard-Vessiot
extension on the other hand. Finally, we compare our Galois theory with the
Galois theory for purely inseparable field extensions.Comment: 37 pages; v2->v3: more cross references to other papers are added in
this version, the introduction is more detailed v3->v4: proof of Thm. 11.5iv)
and of the following corollaries changed and hyperref adde
Conceptual Issues for Noncommutative Gravity on Algebras and Finite Sets
We discuss some of the issues to be addressed in arriving at a definitive
noncommutative Riemannian geometry that generalises conventional geometry both
to the quantum domain and to the discrete domain. This also provides an
introduction to our 1997 formulation based on quantum group frame bundles. We
outline now the local formulae with general differential calculus both on the
base `quantum manifold' and on the structure group Gauge transforms with
nonuniversal calculi, Dirac operator, Levi-Civita condition, Ricci tensor and
other topics are also covered. As an application we outline an intrinsic or
relative theory of quantum measurement and propose it as a possible framework
to explore the link between gravity in quantum systems and entropy.Comment: 17 pages, to appear Proc. Euroconference on Noncommutative Geometry
and Hopf Algebras in Field Theory and Particle Physics, Torino, 1999 -- this
intro for theoretical physicists (mathematicians, see long paper
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