Graphical calculus of Hopf crossed modules

Year 2020, , 695 - 707, 02.04.2020

Kadir Emir

https://doi.org/10.15672/hujms.467966

Cited By: 1

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.

Keywords

Globular, Hopf crossed module, symmetric monoidal category, homotopy

References

• [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: http://globular.science/1610.001v2.
• [8] K. Emir, Globular: Hopf crossed modules, available at: http://globular.science/1611.002v1.
• [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. W. A. Benjamin, 1969.
• [23] X. Tang, A. Weinstein, and C. Zhu, Hopfish algebras, Pacific J. Math. 231 (1), 193–216, 2007.
• [24] J. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42, 409– 428, 1941.