Yıl 2020, Cilt 49 , Sayı 2, Sayfalar 695 - 707 2020-04-02

Graphical calculus of Hopf crossed modules

Kadir EMİR [1]


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.
Globular, Hopf crossed module, symmetric monoidal category, homotopy
  • [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.
Birincil Dil en
Konular Matematik
Bölüm Matematik
Yazarlar

Orcid: 0000-0003-4369-3508
Yazar: Kadir EMİR
Kurum: Depatment of Mathematics and Statistics, Masaryk University
Ülke: Czech Republic


Tarihler

Yayımlanma Tarihi : 2 Nisan 2020

Bibtex @araştırma makalesi { hujms467966, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2020}, volume = {49}, pages = {695 - 707}, doi = {10.15672/hujms.467966}, title = {Graphical calculus of Hopf crossed modules}, key = {cite}, author = {EMİR, Kadir} }
APA EMİR, K . (2020). Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics , 49 (2) , 695-707 . DOI: 10.15672/hujms.467966
MLA EMİR, K . "Graphical calculus of Hopf crossed modules". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 695-707 <https://dergipark.org.tr/tr/pub/hujms/issue/53568/467966>
Chicago EMİR, K . "Graphical calculus of Hopf crossed modules". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 695-707
RIS TY - JOUR T1 - Graphical calculus of Hopf crossed modules AU - Kadir EMİR Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.467966 DO - 10.15672/hujms.467966 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 695 EP - 707 VL - 49 IS - 2 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.467966 UR - https://doi.org/10.15672/hujms.467966 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Graphical calculus of Hopf crossed modules %A Kadir EMİR %T Graphical calculus of Hopf crossed modules %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 2 %R doi: 10.15672/hujms.467966 %U 10.15672/hujms.467966
ISNAD EMİR, Kadir . "Graphical calculus of Hopf crossed modules". Hacettepe Journal of Mathematics and Statistics 49 / 2 (Nisan 2020): 695-707 . https://doi.org/10.15672/hujms.467966
AMA EMİR K . Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 695-707.
Vancouver EMİR K . Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 707-695.