Abstract
William M. Singer's theory of extensions of connected Hopf algebras is used to give a complete list of the cocommutative connected Hopf algebras over a field of positive characteristic p which have vector space dimension less than or equal to p 3. The theory shows that there are exactly two noncommutative nonprimitively generated Hopf algebras on the list, one of which is the Hopf algebra corresponding to the sub-Hopf algebra of the Steenrod algebra generated by P 1 and P p . The commutative Hopf algebras are found using Borel's theorem and the primitively generated Hopf algebras using restricted Lie algebras.
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