Rank functions on \((d+2)\)-angulated categories -- a functorial approach

David Nkansah

Rank functions on \((d+2)\)-angulated categories -- a functorial approach

David Nkansah

Department of Mathematics

Research output: Working paper/Preprint › Preprint

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Abstract

We introduce the notion of a rank function on a \((d+2)\)-angulated category \(\CC\) which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for \(d\) an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in \(\CC\) and rank functions defined on morphisms in \(\CC\). Inspired by work of Conde, Gorsky, Marks and Zvonareva, for \(d\) an odd positive integer, we show there is a bijective correspondence between rank functions on \(\proj A\) and additive functions on \(\mod(\proj A)\), where \(\proj A\) is endowed with the Amiot-Lin \((d+2)\)-angulated category structure. This allows us to show that every integral rank function on \(\proj A\) can be decomposed into irreducible rank functions.

Original language	English
Number of pages	24
Publication status	Published - 29 May 2024

Keywords

math.RT
math.CT
math.RA
18G99, 18A25 (primary), 18G80, 18E10, 18G25 (secondary)

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2405.19042v1

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abstract = " We introduce the notion of a rank function on a \((d+2)\)-angulated category \(\CC\) which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for \(d\) an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in \(\CC\) and rank functions defined on morphisms in \(\CC\). Inspired by work of Conde, Gorsky, Marks and Zvonareva, for \(d\) an odd positive integer, we show there is a bijective correspondence between rank functions on \(\proj A\) and additive functions on \(\mod(\proj A)\), where \(\proj A\) is endowed with the Amiot-Lin \((d+2)\)-angulated category structure. This allows us to show that every integral rank function on \(\proj A\) can be decomposed into irreducible rank functions. ",

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N2 - We introduce the notion of a rank function on a \((d+2)\)-angulated category \(\CC\) which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for \(d\) an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in \(\CC\) and rank functions defined on morphisms in \(\CC\). Inspired by work of Conde, Gorsky, Marks and Zvonareva, for \(d\) an odd positive integer, we show there is a bijective correspondence between rank functions on \(\proj A\) and additive functions on \(\mod(\proj A)\), where \(\proj A\) is endowed with the Amiot-Lin \((d+2)\)-angulated category structure. This allows us to show that every integral rank function on \(\proj A\) can be decomposed into irreducible rank functions.

AB - We introduce the notion of a rank function on a \((d+2)\)-angulated category \(\CC\) which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for \(d\) an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in \(\CC\) and rank functions defined on morphisms in \(\CC\). Inspired by work of Conde, Gorsky, Marks and Zvonareva, for \(d\) an odd positive integer, we show there is a bijective correspondence between rank functions on \(\proj A\) and additive functions on \(\mod(\proj A)\), where \(\proj A\) is endowed with the Amiot-Lin \((d+2)\)-angulated category structure. This allows us to show that every integral rank function on \(\proj A\) can be decomposed into irreducible rank functions.

KW - math.RT

KW - math.CT

KW - math.RA

KW - 18G99, 18A25 (primary), 18G80, 18E10, 18G25 (secondary)

M3 - Preprint

BT - Rank functions on \((d+2)\)-angulated categories -- a functorial approach

ER -

Rank functions on \((d+2)\)-angulated categories -- a functorial approach

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Keywords

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