SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2020 / Cilt: 49 - Sayı: 2

Graphical calculus of Hopf crossed modules

Kadir Emir
Pages
695–707
Publication Date
2022-10-18
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

As an application, we introduce the homotopy of Hopf crossed module maps via Globular, and give some of its functorial relations.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We give the graphical notion of crossed modules of Hopf algebras-will be called Hopf crossed modules for short- in a symmetric monoidal category.
📊
Key Findings & Data:We use the web proof assistant Globular to visualize our (colored) string diagrams.
💡
Conclusion & Impact:As an application, we introduce the homotopy of Hopf crossed module maps via Globular, and give some of its functorial relations.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Globular · Hopf crossed module · symmetric monoidal category

Abstract

We give the graphical notion of crossed modules of Hopf algebras-will be called Hopf crossed modules for short- in a symmetric monoidal category. We use the web proof assistant Globular to visualize our (colored) string diagrams. As an application, we introduce the homotopy of Hopf crossed module maps via Globular, and give some of its functorial relations.

🔖

Cite This Article

Emir, K. (2022). Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 2, 695–707. https://ajindex.com/articles/529650

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (22)

  1. K. Bar, A. Kissinger, and J. Vicary, Globular: an online proof assistant for higherdimensional rewriting, Log. Methods Comput. Sci. 14 (1), 2018
  2. D. Bulacu, S. Caenepeel, F. Panaite, and F. Van Oystaeyen, Quasi-Hopf algebras. A categorical approach, Cambridge University Press, 2019
  3. M. Crossley and N. Turgay, Conjugation invariants in the Leibniz-Hopf algebra, J. Pure Appl. Algebra, 217 (12), 2247–2254, 2013
  4. M. Crossley and N. Turgay, Conjugation invariants in the mod 2 dual Leibniz-Hopf algebra, Comm. Algebra 41 (9), 3261–3266, 2013
  5. M. Crossley and S. Whitehouse, On conjugation invariants in the dual Steenrod algebra, Proc. Amer. Math. Soc. 128 (9), 2809–2818, 2000
  6. V. Drinfel’d, Quasi-Hopf algebras, Leningr. Math. J. 1 (6), 1419–1457, 1990
  7. K. Emir, Globular: Homotopy of Hopf crossed module maps, available at:
  8. K. Emir, Globular: Hopf crossed modules, available at:
  9. K. Emir and S. Çetin, From simplicial homotopy to crossed module homotopy in modified categories of interest, Georgian Math. J. doi: 10.1515/gmj-2018-0069
  10. J. Faria Martins, The fundamental 2-crossed complex of a reduced CW-complex, Homology Homotopy Appl. 13 (2), 129–157, 2011
  11. J. Faria Martins, Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy, J. Geom. Phys. 99, 68–110, 2016
  12. M. Hazewinkel, The Leibniz-Hopf algebra and Lyndon words, CWI Report: AM-R 9612, January 1996
  13. A. Joyal and R. Street, Braided monoidal categories, Macquarie Math Reports 860081, November 1986
  14. J. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (2), 179–202, 1982
  15. S. Majid, Strict quantum 2-groups, arXiv:1208.6265
  16. S. Majid, Algebras and Hopf algebras in braided categories, in: Advances in Hopf algebras, 55–105, New York: Marcel Dekker, 1994
  17. S. Majid, Foundations of quantum group theory, Cambridge University Press, 1995
  18. S. Majid, What is. a quantum group?, Notices Amer. Math. Soc. 53 (1), 30–31, 2006
  19. J. Milnor, The Steenrod algebra and its dual, Ann. Math. (2) 67, 150–171, 1958
  20. D. Radford, The structure of hopf algebras with a projection, J. Algebra 92 (2), 322– 347, 1985
  21. P. Selinger, A survey of graphical languages for monoidal categories. in: New structures for physics, 289–355. Berlin: Springer, 2011
  22. M. Sweedler, Hopf algebras, Mathematics lecture note series.