Frobenius-Perron theory for projective schemes

Abstract

The Frobenius-Perron theory of an endofunctor of a k-linear category (recently introduced in [CG]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes.In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras.

Examples of non-semisimple Hopf algebra actions on Artin-Schelter regular algebras

“…Since then it has become an extremely useful invariant in the study of fusion categories and representations of semisimple (weak and/or quasi-)Hopf algebras. Recently, a new definition of Frobenius-Perron dimension was introduced in [CGW1,CGW2] where the original definition was extended from an object in a semisimple finite tensor category to an endofunctor of any k-linear category. In particular, it is defined for objects in non-semisimple k-linear monoidal categories.…”

“…In particular, it is defined for objects in non-semisimple k-linear monoidal categories. This new Frobenius-Perron dimension has been computed in various cases, see [CGW1,CGW2,Xu,ZWD,ZZ].…”

Frobenius-Perron theory of representation-directed algebras

“…It can be viewed as a generalization of the Frobenius-Perron dimension of an object in a fusion category introduced by Etingof-Nikshych-Ostrik [8]. It was shown in [3,4,17] that the Frobenius-Perron dimension has strong connections with the representation type of a category.…”