A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

The hit problem for a cohomology module over the Steenrod algebra $A$ asks for a minimal set of $A$ –generators for the module. In this paper we consider the symmetric algebras over the field $F_{p}$ , for $p$ an arbitrary prime, and treat the equivalent problem of determining the set of $A^{*}$ –primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action of the Kudo–Araki–May algebra $K$ , and consider the $K$ –module structure of the primitives, which form a sub $K$ –algebra of the dual of the infinite symmetric algebra. Our examples show that the $K$ –action on the primitives is not free. Our new action encompasses, on the finite symmetric algebras, the operators introduced by Kameko for studying the hit problem.

PDF Access Denied

Warning: We have not been able to recognize your IP address 47.88.87.18 as that of a subscriber to this journal. Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form.

Or, visit our subscription page for instructions on purchasing a subscription. You may also contact us at contact@msp.org or by using our contact form.

Or, you may purchase this single article for USD 29.95:

Keywords

hit problem, symmetric algebra, Steenrod algebra, Kudo–Araki–May algebra, Dyer–Lashof algebra, $\mathit{BO}$, $\mathit{BU}$

Mathematical Subject Classification 2000

Primary: 16T05, 16T10, 16W22, 55R45, 55S10, 55S12

Secondary: 16W50, 55R40, 57T05, 57T10, 57T25

References

Publication

Received: 10 July 2010

Revised: 9 February 2011

Accepted: 6 March 2011

Published: 3 June 2011

Authors

David Pengelley
New Mexico State University Las Cruces NM 88003 USA
http://www.math.nmsu.edu/~davidp
Frank Williams
New Mexico State University Las Cruces NM 88003 USA

Volume 11, issue 3 (2011)

David Pengelley and Frank Williams

Abstract