Lots of Hopf Algebras, I

The purpose of this paper is to give definitions of some new Hopf algebras analogous to the mod-p Steenrod algebras. For example, we produce a Hopf algebra over the integers whose reduction mod-p is the mod-p Steenrod algebra. Just for fun, we “quantize” these algebras too! Further study of the examples of this paper will be done in part II. 1 Deformations of Hopf Algebras Suppose that k is a commutative ring with 1, and that (A, ι, μ, ,∆, S) is a Hopf algebra over k. Here, ι is the unit, the counit, μ the multiplication, ∆ the comultiplication and S the antipode of A. An n − parameter deformation of A is a topological Hopf algebra (A~h, ι, μ~h, ,∆~h, S~h) over the power series ring k[[ ~h]] (where ~h = (h1, . . . , hn) is a set of n commuting indeterminates) such that a) A~h is isomorphic to A[[ ~h]] as a topological k[[~h]]-module. b) μ~h ≡ μ mod ~h, ∆~h ≡ ∆ mod ~h. Two deformations A~h and B~h of the same Hopf algebra A are equivalent if there is an isomorphism f~h : A~h → B~h of Hopf algebras over k[[~h]] which is the identity mod ~h. (We are using the notations and definitions of [2].) We are assuming that the unit and counit of A~h are obtained by simply extending those of A, so that there will be no difference in the notations for these maps, whether considered on A or on A~h. Note that ∆~h : A[[ ~h]]→ A[[~h]] ⊗̂k[[~h]]A[[h]] and that μ~h : A[[ ~h]] ⊗̂k[[~h]]A[[h]]→ A[[~h]]; the completion indicated by “⊗̂” is the ~h-adic completion. We will also make the natural identification of k[[~h]]-modules A[[~h]] ⊗̂k[[~h]]A[[h]] ∼= (A⊗k A)[[~h]]