Hopf algebra forms of the multiplicative group and other groups

Abstract

The multiplicative group functor, which associates with each k-algebra its group of units, is affine with Hopf algebra k[x,x⁻¹]. The purpose of this paper is to determine explicitly all Hopf algebra forms of k[x,x⁻¹] with only minor restrictions on k (2 not a zero-divisor and Pic₍₂₎(k)=0). We also describe explicitly (by generators and relations) the Hopf algebra forms of kC₃, kC₄ and kC₆, where C_n is the cyclic group of order n. Some of our results could be drawn from [1,III §5.3.3] where a similar result as ours is indicated (and left as an exercise). We prefer however a less technical approach, in particular we do not use the extended theory of algebraic groups and functor sheaves.