Research Article
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Year 2020, Volume: 49 Issue: 2, 695 - 707, 02.04.2020
https://doi.org/10.15672/hujms.467966

Abstract

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.

Graphical calculus of Hopf crossed modules

Year 2020, Volume: 49 Issue: 2, 695 - 707, 02.04.2020
https://doi.org/10.15672/hujms.467966

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.

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.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Kadir Emir 0000-0003-4369-3508

Publication Date April 2, 2020
Published in Issue Year 2020 Volume: 49 Issue: 2

Cite

APA Emir, K. (2020). Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics, 49(2), 695-707. https://doi.org/10.15672/hujms.467966
AMA Emir K. Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics. April 2020;49(2):695-707. doi:10.15672/hujms.467966
Chicago Emir, Kadir. “Graphical Calculus of Hopf Crossed Modules”. Hacettepe Journal of Mathematics and Statistics 49, no. 2 (April 2020): 695-707. https://doi.org/10.15672/hujms.467966.
EndNote Emir K (April 1, 2020) Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics 49 2 695–707.
IEEE K. Emir, “Graphical calculus of Hopf crossed modules”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, pp. 695–707, 2020, doi: 10.15672/hujms.467966.
ISNAD Emir, Kadir. “Graphical Calculus of Hopf Crossed Modules”. Hacettepe Journal of Mathematics and Statistics 49/2 (April 2020), 695-707. https://doi.org/10.15672/hujms.467966.
JAMA Emir K. Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics. 2020;49:695–707.
MLA Emir, Kadir. “Graphical Calculus of Hopf Crossed Modules”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, 2020, pp. 695-07, doi:10.15672/hujms.467966.
Vancouver Emir K. Graphical calculus of Hopf crossed modules. Hacettepe Journal of Mathematics and Statistics. 2020;49(2):695-707.