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 \#_U (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, which we prove to be an explicitly determined closed subgroup of G. A selection of examples illustrate our results.