Lots of Hopf Algebras

The purpose of this paper is to give a “universal” construction of Hopf algebras over any commutative ring. The mod-pSteenrod algebras are all examples of this construction; other examples of Hopf algebras related to the mod-pSteenrod algebras are also produced. Then we give an argument to produce deformations of these Hopf algebras via “twisting.” The techniques of this paper are, in a way, “scheme-theoretic,” producing many rather complicated Hopf algebras, only a few of which are studied in any detail here; however, we hope to return to the study of some of these Hopf algebras in future work.