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A faithful linear-categorical action of the mapping class group of a surface with boundary

Robert Lipshitz, Peter Ozsváth, Dylan P. Thurston (2013)

Journal of the European Mathematical Society

We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin c -structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology.

A note on model structures on arbitrary Frobenius categories

Zhi-wei Li (2017)

Czechoslovak Mathematical Journal

We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category such that the homotopy category of this model structure is equivalent to the stable category ̲ as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact (closed)...

Affine braid group actions on derived categories of Springer resolutions

Roman Bezrukavnikov, Simon Riche (2012)

Annales scientifiques de l'École Normale Supérieure

In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a “categorical version” of Kazhdan-Lusztig-Ginzburg’s construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course...

An explicit construction for the Happel functor

M. Barot, O. Mendoza (2006)

Colloquium Mathematicae

An easy explicit construction is given for a full and faithful functor from the bounded derived category of modules over an associative algebra A to the stable category of the repetitive algebra of A. This construction simplifies the one given by Happel.

Catégories dérivées et géométrie birationnelle

Raphaël Rouquier (2004/2005)

Séminaire Bourbaki

À l’origine conçue comme un outil technique, la catégorie dérivée des faisceaux cohérents d’une variété algébrique est apparue lors de ces dix dernières années comme un invariant important dans l’étude birationnelle des variétés algébriques. Des problèmes d’invariance birationnelle et de minimisation de la catégorie dérivée sont apparus, inspirés par la conjecture homologique de symétrie miroir de Kontsevich et le programme de Mori de modèles minimaux pour les variétés algébriques. Nous présenterons...

Categorification of the virtual braid groups

Anne-Laure Thiel (2011)

Annales mathématiques Blaise Pascal

We extend Rouquier’s categorification of the braid groups by complexes of Soergel bimodules to the virtual braid groups.

Classification of discrete derived categories

Grzegorz Bobiński, Christof Geiß, Andrzej Skowroński (2004)

Open Mathematics

The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.

Cluster categories for algebras of global dimension 2 and quivers with potential

Claire Amiot (2009)

Annales de l’institut Fourier

Let k be a field and A a finite-dimensional k -algebra of global dimension 2 . We construct a triangulated category 𝒞 A associated to A which, if  A is hereditary, is triangle equivalent to the cluster category of A . When 𝒞 A is Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. Our results also...

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