Twisted unipotent groups

We study the algebraic structure and representation theory of the Hopf algebras JO(G)J when G is an affine algebraic unipotent group over C with dim(G) = n and J is a Hopf $2$-cocycle for G. The cotriangular Hopf algebras JO(G)J have the same coalgebra structure as O(G) but a deformed multiplication. We show that they are involutive n-step iterated Hopf Ore extensions of derivation type. The 2-cocycle J has as support a closed subgroup T of G, and JO(G)J is a crossed product S #σU(t), where t is the Lie algebra of T and S is a deformed coideal subalgebra. The simple JO(G)J-modules are stratified by a family of factor algebras JO(Zg)J, parametrised by the double cosets $TgT$ of T in G. The finite dimensional simple JO(G)J-modules are all 1-dimensional, so form a group Γ, which we prove to be an explicitly determined closed subgroup of G. A selection of examples illustrate our results.