Frobenius–Perron theory of endofunctors

Chen, Jianmin; Gao, Zhibin; Wicks, Elizabeth; Zhang, James J.; Zhang, Xiao­hong; Zhu, Hong

doi:10.2140/ant.2019.13.2005

Cited by 13 publications

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“…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.…”

Section: Comments Projects and Remarksmentioning

confidence: 99%

“…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].…”

Section: Comments Projects and Remarksmentioning

confidence: 99%

“…Sometime it is possible to work out all brick modules which are a special class of indecomposable module. A left B-module is called a brick if Hom B (M, M ) = k. Brick modules are fundamental objects in the study of Frobenius-Perron dimension of endofunctors [CGW1,ZZ]. Even when B is of wild representation type, it would be extremely helpful to understand all brick modules.…”

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

Chen¹,

Wang²,

Zhang³

2020

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Let k be a base field of characteristic p > 0 and let U be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful U -actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.Recall that a Hopf H-action on an algebra A is called inner-faithful if there is no nonzero Hopf ideal I ⊆ H such that IA = 0 [CKWZ1, Definition 1.5]. Our goal is to construct examples of inner-faithful and homogeneous U -actions on T where U is the non-semisimple Hopf algebra given in Definition 0.1 and T is a connected graded Artin-Schelter regular algebra. The main result consists of Proposition 3.2 and Theorem 0.5 that together classify all inner-faithful U -actions on noetherian Koszul Artin-Schelter regular algebras of global dimension at most three.Throughout let k be a base field with char k = p > 0.Definition 0.1. Let U be the k-algebra generated by u and w and subject to the relations (E0.1.1) u p = 0, w p = w, [w, u](:= wu − uw) = u.

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Section: Comments Projects and Remarksmentioning

confidence: 99%

“…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].…”

Section: Comments Projects and Remarksmentioning

confidence: 99%

mentioning

confidence: 99%

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

Chen¹,

Wang²,

Zhang³

2020

Preprint

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“…Recently, the Frobenius-Perron dimension of an endofunctor of a category was introduced by the authors in [3]. 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].…”

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confidence: 99%

Frobenius-Perron theory of representation-directed algebras

Chen¹,

Chen²

2022

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We study the Frobenius-Perron dimension of representation-directed algebras and quotient algebras of canonical algebras of type ADE, prove that the Frobenius-Perron dimension of a representation-directed algebra is always zero and the Frobenius-Perron dimension of a quotient algebra of canonical algebras of type ADE is 0 or 1. Moreover, we give a sufficient and necessary condition for a quotient algebra of a canonical algebra of type ADE under which its Frobenius-Perron dimension is 0.

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“…It was first defined for commutative fusion rings by Fröhlich and Kerler [14] as functions satisfying certain properties. The theory of Frobenius-Perron dimensions for general fusion rings and categories was developed by Etingof, Nikshych and Ostrick [11]; for some other cases, we refer to [9,12,5,10,22] and references therein. The key ingredient here is to define a function on a nice Z + -ring of finite rank whose value at a basis element is the spectral radius of the induced linear operator as guaranteed by the Frobenius-Perron theory.…”

Section: Introductionmentioning

confidence: 99%

On Frobenius-Perron Dimension

Li¹,

Shifler²,

Yang³

et al. 2019

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We propose a notion of Frobenius-Perron dimension for certain free Z-modules of infinite rank and compute it for the Z-modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. The definition of Frobenius-Perron dimension that we are introducing naturally generalizes the well-known Frobenius-Perron dimension on the category of finite dimensional complex representations of a finite group.

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Frobenius–Perron theory of endofunctors

Abstract: We introduce the Frobenius-Perron dimension of an endofunctor of a k-linear category and provide some applications.

Cited by 13 publications

References 55 publications

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

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

Frobenius-Perron theory of representation-directed algebras

On Frobenius-Perron Dimension

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