22,022 research outputs found
Compactification of M(atrix) theory on noncommutative toroidal orbifolds
It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori
arise naturally in consideration of toroidal compactifications of M(atrix)
theory. A similar analysis of toroidal Z_{2} orbifolds leads to the algebra
B_{\theta} that can be defined as a crossed product of noncommutative torus and
the group Z_{2}. Our paper is devoted to the study of projective modules over
B_{\theta} (Z_{2}-equivariant projective modules over a noncommutative torus).
We analyze the Morita equivalence (duality) for B_{\theta} algebras working out
the two-dimensional case in detail.Comment: 19 pages, Latex; v2: comments clarifying the duality group structure
added, section 5 extended, minor improvements all over the tex
Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories
Auslander-Reiten theory is fundamental to study categories which appear in
representation theory, for example, modules over artin algebras, Cohen-Macaulay
modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves
on projective curves. In these Auslander-Reiten theories, the number `2' is
quite symbolic. For one thing, almost split sequences give minimal projective
resolutions of simple functors of projective dimension `2'. For another,
Cohen-Macaulay rings of Krull-dimension `2' provide us with one of the most
beautiful situation in representation theory, which is closely related to
McKay's observation on simple singularities. In this sense, usual
Auslander-Reiten theory should be `2-dimensional' theory, and it be natural to
find a setting for higher dimensional Auslander-Reiten theory from the
viewpoint of representation theory and non-commutative algebraic geometry.
We introduce maximal -orthogonal subcategories as a natural domain of
higher dimensional Auslander-Reiten theory which should be
`-dimensional'. We show that the -Auslander-Reiten translation
functor and the -Auslander-Reiten duality can be defined quite naturally for
such categories. Using them, we show that our categories have {\it -almost
split sequences}, which give minimal projective resolutions of simple objects
of projective dimension `' in functor categories. We show that an
invariant subring (of Krull-dimension `') corresponding to a finite
subgroup of has a natural maximal -orthogonal
subcategory. We give a classification of all maximal 1-orthogonal subcategories
for representation-finite selfinjective algebras and representation-finite
Gorenstein orders of classical type.Comment: 25 pages. Final Version. To appear in Adv. Mat
Projective Fourier Duality and Weyl Quantization
The Weyl-Wigner correspondence prescription, which makes large use of Fourier
duality, is reexamined from the point of view of Kac algebras, the most general
background for noncommutative Fourier analysis allowing for that property. It
is shown how the standard Kac structure has to be extended in order to
accommodate the physical requirements. An Abelian and a symmetric projective
Kac algebras are shown to provide, in close parallel to the standard case, a
new dual framework and a well-defined notion of projective Fourier duality for
the group of translations on the plane. The Weyl formula arises naturally as an
irreducible component of the duality mapping between these projective algebras.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 102Kb, 44 pages, no figures.
requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with text
improvements and crucial typos correction
On a common generalization of Koszul duality and tilting equivalence
We propose a new definition of Koszulity for graded algebras where the degree
zero part has finite global dimension, but is not necessarily semi-simple. The
standard Koszul duality theorems hold in this setting. We give an application
to algebras arising from multiplicity free blocks of the BGG category
Properly stratified algebras and tilting
We study the properties of tilting modules in the context of properly
stratified algebras. In particular, we answer the question when the Ringel dual
of a properly stratified algebra is properly stratified itself, and show that
the class of properly stratified algebras for which the characteristic tilting
and cotilting modules coincide is closed under taking the Ringel dual. Studying
stratified algebras, whose Ringel dual is properly stratified, we discover a
new Ringel-type duality for such algebras, which we call the two-step duality.
This duality arises from the existence of a new (generalized) tilting module
for stratified algebras with properly stratified Ringel dual. We show that this
new tilting module has a lot of interesting properties, for instance, its
projective dimension equals the projectively defined finitistic dimension of
the original algebra, it guarantees that the category of modules of finite
projective dimension is contravariantly finite, and, finally, it allows one to
compute the finitistic dimension of the original algebra in terms of the
projective dimension of the characteristic tilting module.Comment: A revised version of the preprint 2003:31, Department of Mathematics,
Uppsala Universit
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