Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 14, 2018

The Arnon bases in the Steenrod algebra

  • Neşet Deniz Turgay ORCID logo EMAIL logo and Ismet Karaca

Abstract

Let 𝒜=𝒜p be the modp Steenrod algebra, where p is a fixed prime and let 𝒜 denote the Bockstein-free part of 𝒜 at odd primes. Being a connected graded Hopf algebra, 𝒜 has the canonical conjugation χ. Using this map, we introduce a relationship between the X- and Z-bases of 𝒜. We show that these bases restrict to give bases to the well-known sub-Hopf algebras 𝒜(n-1), n1, of 𝒜.

MSC 2010: 55S10; 55S05; 57T05

Acknowledgements

The authors would like to express their deep gratitude to the reviewer for his/her valuable comments and helpful suggestions for the improvement of this work.

References

[1] D. Arnon, Monomial bases in the Steenrod algebra, J. Pure Appl. Algebra 96 (1994), no. 3, 215–223. 10.1016/0022-4049(94)90099-XSearch in Google Scholar

[2] D. M. Davis, The antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc. 44 (1974), 235–236. 10.1090/S0002-9939-1974-0328934-1Search in Google Scholar

[3] D. Y. Emelyanov and Th. Yu. Popelensky, On monomial bases in the mod p Steenrod algebra, JP J. Fixed Point Theory Appl. 17 (2015), no. 2, 341–353. 10.1007/s11784-014-0166-3Search in Google Scholar

[4] V. Giambalvo and H. R. Miller, More on the anti-automorphism of the Steenrod algebra, Algebr. Geom. Topol. 11 (2011), no. 5, 2579–2585. 10.2140/agt.2011.11.2579Search in Google Scholar

[5] I. Karaca, Monomial bases in the mod-p Steenrod algebra, Czechoslovak Math. J. 55(130) (2005), no. 3, 699–707. 10.1007/s10587-005-0057-2Search in Google Scholar

[6] I. Karaca and I. Y. Karaca, On conjugation in the mod-p Steenrod algebra, Turkish J. Math. 24 (2000), no. 4, 359–365. Search in Google Scholar

[7] J. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. 10.1142/9789814401319_0006Search in Google Scholar

[8] J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–MacLane, Comment. Math. Helv. 27 (1953), 198–232. 10.1007/BF02564562Search in Google Scholar

[9] J. H. Silverman, Conjugation and excess in the Steenrod algebra, Proc. Amer. Math. Soc. 119 (1993), no. 2, 657–661. 10.1090/S0002-9939-1993-1152292-8Search in Google Scholar

[10] N. E. Steenrod and D. B. A. Epstein, Cohomology Operations, Ann. of Math. Stud. 50, Princeton University Press, Princeton, 1962. Search in Google Scholar

[11] P. D. Straffin. Jr., Identities for conjugation in the Steenrod algebra, Proc. Amer. Math. Soc. 49 (1975), 253–255. 10.1090/S0002-9939-1975-0380796-3Search in Google Scholar

[12] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. 10.1007/BF02566923Search in Google Scholar

[13] N. D. Turgay, A remark on the conjugation in the Steenrod algebra, Commun. Korean Math. Soc. 30 (2015), no. 3, 269–276. 10.4134/CKMS.2015.30.3.269Search in Google Scholar

[14] G. Walker and R. M. W. Wood, The nilpotence height of Sq2n, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1291–1295. 10.1090/S0002-9939-96-03203-0Search in Google Scholar

[15] G. Walker and R. M. W. Wood, The nilpotence height of Ppn, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 1, 85–93. 10.1017/S0305004197001813Search in Google Scholar

[16] C. T. C. Wall, Generators and relations for the Steenrod algebra, Ann. of Math. (2) 72 (1960), 429–444. 10.2307/1970225Search in Google Scholar

[17] R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. Lond. Math. Soc. 27 (1995), no. 4, 380–386. 10.1112/blms/27.4.380Search in Google Scholar

[18] R. M. W. Wood, Problems in the Steenrod algebra, Bull. Lond. Math. Soc. 30 (1998), no. 5, 449–517. 10.1112/S002460939800486XSearch in Google Scholar

Received: 2016-09-16
Revised: 2016-11-23
Accepted: 2016-12-05
Published Online: 2018-11-14
Published in Print: 2020-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 19.4.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2018-0076/html
Scroll to top button