We introduce the continuous Frobenius category. This category is constructed using representations of the circle over a discrete valuation ring. We show that it is Krull-Schmidt with one indecomposable object for each pair of not necessarily distinct points on the circle. By putting restrictions on these points we obtain various subquotient categories with good properties. The main purpose of o...

For a Calabi-Yau triangulated category C of Calabi-Yau dimension d with a d−cluster tilting subcategory T , the decomposition of C is determined by the decomposition of T satisfying ”vanishing condition” of negative extension groups, namely, C = ⊕i∈ICi, where Ci, i ∈ I are triangulated subcategories, if and only if T = ⊕i∈ITi, where Ti, i ∈ I are subcategories with HomC(Ti[t],T j) = 0,∀1 ≤ t ≤ ...

The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived category of right-modules.

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

We discuss an example of a triangulated Hopf category related to SL(2). It is an equivariant derived category equipped with multiplication and comultiplication functors and structure isomorphisms. We prove some coherence equations for structure isomorphisms. In particular, the Hopf category is monoidal.

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct an action of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models It seems that two different types of categories can be associated with singularities (or singularities of maps). Categories of the firs...

We introduce and develop an analogous of the AuslanderBuchweitz approximation theory (see [2]) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category T , which are based on the behavior of certain subcategories under finiteness of resolutions and vanishin...