The cohomology of rank three Hopf algebras
Abstract
The rank three graded connected cocommutative Hopf algebras over a perfect field are classified and the cohomology algebras with the Steenrod algebra actions are computed. Using this data, the version of Quillen's F-isomorphism theorem for finite graded connected cocommutative Hopf algebras which is suggested by work of Clarence Wilkerson and Haynes Miller is studied in the special case of rank three Hopf algebras. William M. Singer's theory of extensions of Hopf algebras is used to accomplish the classification, although his cohomology groups are not computed explicitly. The classification shows that the expected Hopf algebras arise plus one additional, which is noncommutative and primitively generated, in characteristic two. The expected ones are: eight which are commutative, two noncommutative primitively generated families corresponding to the two families of p-groups of order p$\sp3,$ and the noncommutative nonprimitively generated sub-Hopf algebra of the Steenrod algebra which is generated by P$\sp1$ and P$\sp{\rm p}.$ The cohomology algebras are computed using the change of rings spectral sequence and the action of the Steenrod algebra on the cobar construction. There are several nontrivial extension problems to be solved in order to complete this computation. Using this information, the Quillen category for each of the rank three Hopf algebras is determined and the map to the inverse limit calculated explicitly. The generalization of Quillen's F-isomorphism theorem is seen to hold for these Hopf algebras.
Degree
Ph.D.
Advisors
Wilkerson, Purdue University.
Subject Area
Mathematics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.