Copure semisimple categories and almost split maps

Strongly copure projective objects in triangulated categories

In this paper, we introduce and investigate the notions of ξ-strongly copure projective objects in a triangulated category. This extends Asadollahi’s notion of ξ-Gorenstein projective objects. Then we study the ξ-strongly copure projective dimension and investigate the existence of ξ-strongly copure projective precover.

Almost semisimple algebras

Finitely semisimple spherical categories and modular categories are self - dual

We show that every essentially small finitely semisimple k-linear additive spherical category in which k = End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as...

A construction of semisimple tensor categories

Let A be an abelian category such that every object has only finitely many subobjects. From A we construct a semisimple tensor category T . We show that T interpolates the categories Rep(Aut(p), K) where p runs through certain projective pro-objects of A. This extends a construction of Deligne for symmetric groups.

Almost Split Morphisms, Preprojective Algebras and Multiplication Maps of Maximal Rank

With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in the commutative polynomial ring. The result follows from a more general theorem about the maximal rank property of a minimal almost split mor...