For an integer w, let Sw be the algebraic triangulated category generated by a w-spherical object. We determine the Picard group of Sw and show that each orbit category of Sw is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. Moreover, we describe the Ringel–Hall Lie algebra associated to Sw in the sense of Peng and Xiao.

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could...

Stability conditions on triangulated categories were introduced by Bridgeland as a ‘continuous’ generalisation of t-structures. The set of locally-finite stability conditions on a triangulated category is a manifold which has been studied intensively. However, there are mainstream triangulated categories whose stability manifold is the empty set. One example is D ( k[X]/(X) ) , the compact deri...

We prove a structure theorem for triangulated Calabi-Yau categories: An algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category iff it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable categor...

For a complex compact manifold X, denote by T the category Dcoh(OX). • This category is a C-triangulated category, • this category is Ext-finite, that is, ⊕