@TECHREPORT{Duflot96lotsof, author = {Jeanne Duflot and That}, title = {Lots of Hopf Algebras}, institution = {}, year = {1996} }

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Abstract

The purpose of this paper is to give denitions of some new Hopf algebras analo-gous 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; ; ; ;; S) is a Hopf algebra over k. Here, is the unit, the counit, the multiplication, the comultiplication and S the antipode of A. An n − parameter deformation of A is a topological Hopf algebra (A~h; ; ~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) ~h 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 denitions 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 dierence in the notations for these maps, whether considered on A or on A~h. Note that