22,022 research outputs found

    Compactification of M(atrix) theory on noncommutative toroidal orbifolds

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    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

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    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 (n−1)(n-1)-orthogonal subcategories as a natural domain of higher dimensional Auslander-Reiten theory which should be `(n+1)(n+1)-dimensional'. We show that the nn-Auslander-Reiten translation functor and the nn-Auslander-Reiten duality can be defined quite naturally for such categories. Using them, we show that our categories have {\it nn-almost split sequences}, which give minimal projective resolutions of simple objects of projective dimension `n+1n+1' in functor categories. We show that an invariant subring (of Krull-dimension `n+1n+1') corresponding to a finite subgroup GG of GL(n+1,k){\rm GL}(n+1,k) has a natural maximal (n−1)(n-1)-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

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    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

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    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 O\mathcal O

    Properly stratified algebras and tilting

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    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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