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2020, Volume 28, Issue 2: 951-959. doi: 10.3934/era.2020050
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On the mod p Steenrod algebra and the Leibniz-Hopf algebra

  • Faculty of Arts and Sciences, Department of Mathematics, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey
  • Received: 01 March 2020 Revised: 01 May 2020

  • Primary: 55S10, 16T05; Secondary: 57T05

  • Let $ p $ be a fixed odd prime. The Bockstein free part of the mod $ p $ Steenrod algebra, $ \mathcal{A}_p $, can be defined as the quotient of the mod $ p $ reduction of the Leibniz Hopf algebra, $ \mathcal{F}_p $. We study the Hopf algebra epimorphism $ \pi\colon \mathcal{F}_p\to \mathcal{A}_p $ to investigate the canonical Hopf algebra conjugation in $ \mathcal{A}_p $ together with the conjugation operation in $ \mathcal{F}_p $. We also give a result about conjugation invariants in the mod 2 dual Leibniz Hopf algebra using its multiplicative algebra structure.

    Citation: Neşet Deniz Turgay. On the mod p Steenrod algebra and the Leibniz-Hopf algebra[J]. Electronic Research Archive, 2020, 28(2): 951-959. doi: 10.3934/era.2020050

    Related Papers:

  • Let $ p $ be a fixed odd prime. The Bockstein free part of the mod $ p $ Steenrod algebra, $ \mathcal{A}_p $, can be defined as the quotient of the mod $ p $ reduction of the Leibniz Hopf algebra, $ \mathcal{F}_p $. We study the Hopf algebra epimorphism $ \pi\colon \mathcal{F}_p\to \mathcal{A}_p $ to investigate the canonical Hopf algebra conjugation in $ \mathcal{A}_p $ together with the conjugation operation in $ \mathcal{F}_p $. We also give a result about conjugation invariants in the mod 2 dual Leibniz Hopf algebra using its multiplicative algebra structure.



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    [1] J. F. Adams, Lectures on generalised cohomology, in Category Theory, Homology Theory and their Applications, III, Lecture Notes in Mathematics, 99, Springer, Berlin, 1969, 1–138. doi: 10.1007/BFb0081960
    [2] Monomial bases in the Steenrod algebra. J. Pure Appl. Algebra (1994) 96: 215-223.
    [3] M. G. Barratt and H. R. Miller, On the anti-automorphism of the Steenrod algebra, in Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math., 12, Amer. Math. Soc., Providence, RI, 1982, 47–52.
    [4] D. Bulacu, S. Caenepeel, F. Panaite and F. Van Oystaeyen, Quasi-Hopf Algebras, Encyclopedia of Mathematics and its Applications, 171, Cambridge University Press, Cambridge, 2019. doi: 10.1017/9781108582780
    [5] On the Adem relations. Topology (1982) 21: 329-332.
    [6] The intersection of the admissible basis and the Milnor basis of the Steenrod algebra. J. Pure Appl. Algebra (1998) 128: 1-10.
    [7] The Steenrod algebra and other copolynomial Hopf algebras. Bull. London Math. Soc. (2000) 32: 609-614.
    [8] Some Hopf algebras of words. Glasg. Math. J. (2006) 48: 575-582.
    [9] Conjugation invariants in the mod 2 dual Leibniz-Hopf algebra. Comm. Algebra (2013) 41: 3261-3266.
    [10] Conjugation invariants in the Leibniz-Hopf algebra. J. Pure Appl. Algebra (2013) 217: 2247-2254.
    [11] On conjugation invariants in the dual Steenrod algebra. Proc. Amer. Math. Soc. (2000) 128: 2809-2818.
    [12] Higher conjugation cohomology in commutative Hopf algebras. Proc. Edinb. Math. Soc. (2) (2001) 44: 19-26.
    [13] The antiautomorphism of the Steenrod algebra. Proc. Amer. Math. Soc. (1974) 44: 235-236.
    [14] Quasi-Hopf algebras. Leningr. Math. J. (1990) 1: 1419-1457.
    [15] On posets and Hopf algebras. Adv. Math. (1996) 119: 1-25.
    [16] On monomial bases in the mod $p$ Steenrod algebra. J. Fixed Point Theory Appl. (2015) 17: 341-353.
    [17] Graphical calculus of Hopf crossed modules. Hacettepe J. Math. Statistics (2020) 49: 695-707.
    [18] Generalized overlapping shuffle algebras. J. Math. Sci. (New York) (2001) 106: 3168-3186.
    [19] The algebra of quasi-symmetric functions is free over the integers. Adv. Math. (2001) 164: 283-300.
    [20] Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. Monodromy and differential equations. Acta Appl. Math. (2003) 75: 55-83.
    [21] Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. II. Acta Appl. Math. (2005) 85: 319-340.
    [22] Explicit polynomial generators for the ring of quasisymmetric functions over the integers. Acta. Appl. Math. (2010) 109: 39-44.
    [23] S. Kaji, A Maple Code for the Dual Leibniz–Hopf Algebra. Available from: http://www.skaji.org/files/Leibniz-Hopf.mw.
    [24] On conjugation in the mod-$p$ Steenrod algebra. Turkish J. Math. (2000) 24: 359-365.
    [25] Monomial bases in the mod-$p$ Steenrod algebra. Czechoslovak Math. J. (2005) 55: 699-707.
    [26] (1995) Foundations of Quantum Group Theory. Cambridge: Cambridge University Press.
    [27] Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra (1995) 177: 967-982.
    [28] The Steenrod algebra and its dual. Ann. of Math. (2) (1958) 67: 150-171.
    [29] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2), 81, (1965), 211–264. doi: 10.2307/1970615
    [30] Change of basis, monomial relations, and the $P_t^{s}$ bases for the Steenrod algebra. J. Pure Appl. Algebra (1998) 125: 235-260.
    [31] K. G. Monks, STEENROD: A Maple package for computing with the Steenrod algebra, 1995.
    [32] Cohomologie modulo 2 des complexes d'Eilenberg-MacLane. Comment. Math. Helv. (1953) 27: 198-232.
    [33] Conjugation and excess in the Steenrod algebra. Proc. Amer. Math. Soc. (1993) 119: 657-661.
    [34] N. E. Steenrod, Cohomology Operations, Annals of Math Studies, 50, Princeton University Press, Princeton, NJ, 1962.
    [35] W. Stein, et al., Sage Mathematics Software (Version 5.4.1), The Sage Development Team, 2012. Available from: http://www.sagemath.org.
    [36] Identities for conjugation in the Steenrod algebra. Proc. Amer. Math. Soc. (1975) 49: 253-255.
    [37] Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. (1954) 28: 17-86.
    [38] On the conjugation invariant problem in the mod $p$ dual Steenrod algebra. Ital. J. Pure Appl. Math. (2015) 34: 151-158.
    [39] A remark on the conjugation in the Steenrod algebra. Commun. Korean Math. Soc. (2015) 30: 269-276.
    [40] An alternative approach to the Adem relations in the mod 2 Steenrod algebra. Turkish J. Mathematics (2014) 38: 924-934.
    [41] An alternative approach to the Adem relations in the mod $p$ Steenrod algebra. C. R. Acad. Bulgare Sci. (2017) 70: 457-466.
    [42] Invariants under decomposition of the conjugation in the mod 2 dual Leibniz-Hopf algebra. Miskolc Math. Notes (2018) 19: 1217-1222.
    [43] The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra. J. Homotopy Relat. Struct. (2017) 12: 727-739.
    [44] N. D. Turgay and I. Karaca, The Arnon bases in the Steenrod algebra, Georgian Math. J., (2018). doi: 10.1515/gmj-2018-0076
    [45] The nilpotence height of ${S}q^{2n}$. Proc. Amer. Math. Soc. (1996) 124: 1291-1295.
    [46] The nilpotence height of ${P}^{p^n}$. Math. Proc. Cambridge Philos. Soc. (1998) 123: 85-93.
    [47] Generators and relations for the Steenrod algebra. Ann. of Math. (2) (1960) 72: 429-444.
    [48] A note on bases and relations in the Steenrod algebra. Bull. London Math. Soc. (1995) 27: 380-386.
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