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EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants

Part of: Differential topology Connections with homological algebra and category theory Categories with structure

Published online by Cambridge University Press:  13 March 2019

Hua-Lin Huang ,
Zheyan Wan  and
Yu Ye
Hua-Lin Huang
Affiliation:
School of Mathematical Sciences, Fujian Province University Key Laboratory of Computational Science, Huaqiao University, Quanzhou362021, China (hualin.huang@hqu.edu.cn)
Zheyan Wan
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei230026, China (wanzy@mail.ustc.edu.cn; yeyu@ustc.edu.cn)
Yu Ye
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei230026, China (wanzy@mail.ustc.edu.cn; yeyu@ustc.edu.cn)
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Abstract

We provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

Keywords

group cocycleGr-categoryDW invariant

MSC classification

Primary: 20J06: Cohomology of groups
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 57R56: Topological quantum field theories
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1937 - 1964
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Copyright
Copyright © Royal Society of Edinburgh 2019

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