Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T03:19:39.254Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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 ([email protected])
Zheyan Wan
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei230026, China ([email protected]; [email protected])
Yu Ye
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei230026, China ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bulacu, D., Caenepeel, S. and Torrecillas, B.. The braided monoidal structures on the category of vector spaces graded by the Klein group. Proc. Edinburgh Math. Soc. 54 (2011), 613–641.CrossRefGoogle Scholar
2Chen, X., Gu, Z.-C., Liu, Z.-X. and Wen, X.-G.. Symmetry-protected topological orders in interacting bosonic systems. Science 338 (2012), 1604–1606.CrossRefGoogle ScholarPubMed
3Chen, X., Gu, Z.-C., Liu, Z.-X. and Wen, X.-G.. Symmetry protected topological orders and the group cohomology of their symmetry group. Phys. Rev. B 87 (2013), 155114.CrossRefGoogle Scholar
4de Wild Propitius, M.. Topological interactions in broken gauge theories. PhD thesis, University of Amsterdam, 1995.Google Scholar
5Dijkgraaf, R. and Witten, E.. Topological gauge theories and group cohomology. Commun. Math. Phys. 129 (1990), 393–429.CrossRefGoogle Scholar
6Eilenberg, S. and MacLane, S.. Cohomology theory of abelian groups and homotopy theory I. Proc. Natl. Acad. Sci. 36 (1950), 443–447.CrossRefGoogle ScholarPubMed
7Eilenberg, S. and MacLane, S.. Cohomology theory of abelian groups and homotopy theory II. Proc. Natl. Acad. Sci. 36 (1950), 657–663.CrossRefGoogle ScholarPubMed
8Etingof, P. and Gelaki, S.. Finite dimensional quasi-Hopf algebras with radical of codimension 2. Math. Res. Lett. 11 (2004), 685–696.CrossRefGoogle Scholar
9Etingof, P. and Gelaki, S.. On radically graded finite-dimensional quasi-Hopf algebras. Moscow Math. J. 5 (2005), 371–378.CrossRefGoogle Scholar
10Freed, D. S.. Higher algebraic structures and quantization. Commun. Math. Phys. 159 (1994), 343–398.CrossRefGoogle Scholar
11Frucht, R.. Über die Darstellung endlicher abelscher Gruppen durch Kollineationen. Journal für die reine und angewandte Mathematik (Crelles Journal) 166 (1932), 16–29.Google Scholar
12Gelaki, S.. Basic quasi-Hopf algebras of dimension n 3. J. Pure Appl. Algebra 198 (2005), 165–174.CrossRefGoogle Scholar
13Hatcher, A.. Algebraic topology (Cambridge: Cambridge University Press, 2002).Google Scholar
14Hoàng, X. S.. Gr-catégories. PhD thesis, Université Paris VII, (1975).Google Scholar
15Hochschild, G. and Serre, J.-P.. Cohomology of group extensions. Trans. Amer. Math. Soc., 74 (1953), 110–134.CrossRefGoogle Scholar
16Hu, Y., Wan, Y. and Wu, Y.-S.. Twisted quantum double model of topological phases in two dimensions. Phys. Rev. B 87 (2013), 125114.CrossRefGoogle Scholar
17Huang, H.-L., Liu, G. and Ye, Y.. Quivers, quasi-quantum groups and finite tensor categories. Commun. Math. Phys. 303 (2011), 595–612.CrossRefGoogle Scholar
18Huang, H.-L., Liu, G. and Ye, Y.. The braided monoidal structures on a class of linear Gr-categories. Algebr. Represent. Th. 17 (2014), 1249–1265.CrossRefGoogle Scholar
19Huang, H.-L., Liu, G., Yang, Y. and Ye, Y.. Finite quasi-quantum groups of rank two. arXiv preprint arXiv:1508.04248 (2015).Google Scholar
20Huang, H.-L., Liu, G., Yang, Y. and Ye, Y.. Finite quasi-quantum groups of diagonal type. Journal für die reine und angewandte Mathematik (Crelles Journal) (2018). doi: 10.1515/crelle-2017-0058.Google Scholar
21Joyal, A. and Street, R.. Braided tensor categories. Adv. Math. 102 (1993), 20–78.CrossRefGoogle Scholar
22Karpilovsky, G.. Projective representations of finite groups, vol. 94, Monographs and Textbooks in Pure and Applied Mathematics (New York: Marcel Dekker Inc, 1985).Google Scholar
23Lyndon, R. C.. The cohomology theory of group extensions. Duke Math. J. 15 (1948), 271–292.CrossRefGoogle Scholar
24Moore, G. and Seiberg, N.. Classical and quantum conformal field theory. Commun. Math. Phys. 123 (1989), 177–254.CrossRefGoogle Scholar
25Turaev, V.. Dijkgraaf–Witten invariants of surfaces and projective representations of groups. J. Geom. Phys. 57 (2007), 2419–2430.CrossRefGoogle Scholar
26Wan, Y., Wang, J. C. and He, H.. Twisted gauge theory model of topological phases in three dimensions. Phys. Rev. B 92 (2015), 045101.CrossRefGoogle Scholar
27Wang, J. C. and Wen, X.-G.. Non-Abelian string and particle braiding in topological order: Modular SL(3, ℤ) representation and (3+1)-dimensional twisted gauge theory. Phys. Rev. B 91 (2015), 035134.Google Scholar
28Wang, J., Ohmori, K., Putrov, P., Zheng, Y., Wan, Z., Guo, M., Lin, H., Gao, P. and Yau, S.-T.. Tunneling topological vacua via extended operators:(Spin-) TQFT spectra and boundary deconfinement in various dimensions. Progr. Theor. Exp. Phys. 2018 (2018), 053A01.CrossRefGoogle Scholar
29Wang, J. C., Gu, Z.-C. and Wen, X.-G.. Field-theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology, and beyond. Phys. Rev. Lett. 114 (2015), 031601.Google ScholarPubMed
30Weibel, C. A.. An introduction to homological algebra, vol. 38, Cambridge Studies in Advanced Mathematics (Cambridge: Cambridge University Press, 1995).Google Scholar
31Wen, X.-G.. Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1 dimensions. Phys. Rev. B 95 (2017), 205142.CrossRefGoogle Scholar