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On Support τ-tilting Modules over Endomorphism Algebras of rigid Ob jects

Acta Mathematica Sinica,English Series 2015年第9期31卷 1508-1516页

作者： Wen CHANG Jie ZHANG Bin ZHU Department of Mathematical Sciences Tsinghua University School of Mathematics and Statistics Beijing institute of technology

We consider a Krull-Schmidt, Hom-finite, 2-Calabi Yau triangulated category with a basic rigid object T, and show a bijection between the set of isomorphism classes of basic rigid objects in the finite presented category pr T of T and the set of isomorphism classes of basic T-rigid pairs in the module category of the endomorphism algebra Endc（T）op. As a consequence, basic maximal objects in prT are one-to-one correspondence to basic support τ-tilting modules over Endc（T）op. This is a generalization of correspondences established by Adachi-Iyama-Reiten.

关键词： rigid object maximal rigid object τ-rigid object finite presented category

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Gaussian-curvature-derived invariants for isometry

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Science China(Information Sciences) 2013年第9期56卷 219-230页

作者： CAO WeiGuo HU Ping LIU YuJie GONG Ming LI Hua Institute of Computing Technology Chinese Academy of Sciences Graduate University of Chinese Academy of Sciences School of Computer Science and Communication Engineering China University of Petroleum

Surface deformations without tearing or stretching, preserving the intrinsic properties, are called isometries. This paper presents a new definition of Gaussian curvature moments (GCMs) by the integral of n power of Gaussian curvature. Then a series of moment invariants, called Gaussian curvature moment invariants (GCMIs), are derived via GCMs. These moment invariants share many good properties under rigid transformations and isometric non-rigid transformations, and Gaussian-Bonnet theorem is a special case of GCMI. GCMIs are invariant under isometry and scaling transformations. We construct an invariant vector as a descriptor for a surface via GCMIs, and a modified χ2 distance is defined as a measure of similarity. Finally, experiments show that it is a reliable descriptor for isometric non-rigid shape.

关键词： rigid object non-rigid object isometry scale Gaussian curvature moment moment invariant

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Auslander-Reiten Sequences or Triangles Related to rigid Subcategories

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Algebra Colloquium 2016年第1期23卷 1-14页

作者： Ming Lu Department of Mathematics Sichuan University Chengdu 610064 China

Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is （2d-1）-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification： 16G20, 16G70

关键词： Calabi-Yau triangulated category cluster category cluster-tilting subcategory functorially finite rigid subcategory rigid object

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