252 research outputs found
Computing homomorphisms between holonomic D-modules
Let K be a subfield of the complex numbers, and let D be the Weyl algebra of
K-linear differential operators on K[x_1,...,x_n]. If M and N are holonomic
left D-modules we present an algorithm that computes explicit generators for
the finite dimensional vector space hom_D(M,N). This enables us to answer
algorithmically whether two given holonomic modules are isomorphic. More
generally, our algorithm can be used to get explicit generators for
ext^i_D(M,N) for any i.Comment: 30 pages, AMS-LaTex, uses
verbatim,amsmath,latexsym,amssymb,amsbsy,diagram
Bernstein-Gelfand-Gelfand sequences
This paper is devoted to the study of geometric structures modeled on
homogeneous spaces G/P, where G is a real or complex semisimple Lie group and
is a parabolic subgroup. We use methods from differential geometry
and very elementary finite-dimensional representation theory to construct
sequences of invariant differential operators for such geometries, both in the
smooth and the holomorphic category. For G simple, these sequences specialize
on the homogeneous model G/P to the celebrated (generalized)
Bernstein-Gelfand-Gelfand resolutions in the holomorphic category, while in the
smooth category we get smooth analogs of these resolutions. In the case of
geometries locally isomorphic to the homogeneous model, we still get
resolutions, whose cohomology is explicitly related to a twisted de Rham
cohomology. In the general (curved) case we get distinguished curved analogs of
all the invariant differential operators occurring in Bernstein-Gelfand-Gelfand
resolutions (and their smooth analogs).
On the way to these results, a significant part of the general theory of
geometrical structures of the type described above is presented here for the
first time.Comment: 45 page
Simple D-module components of local cohomology modules
For a projective variety V in P^n over a field of characteristic zero, with
homogeneous ideal I in A = k[x0,x1,...,xn], we consider the local cohomology
modules H^i_I(A). These have a structure of holonomic D-module over A, and we
investigate their filtration by simple D-modules. In case V is nonsingular, we
can describe completely these simple components in terms of the Betti numbers
of V.Comment: 22 page
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
Polynomial and rational solutions of holonomic systems
The aim of this paper is to give two new algorithms, which are elimination
free, to find polynomial and rational solutions for a given holonomic system
associated to a set of linear differential operators in the Weyl algebra D =
k where k is a subfield of the complex numbers.Comment: 20 page
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