Twisted unipotent groups

We study the algebraic structure and representation theory of the Hopf algebras ${}_{J}O{\left(G\right)}_J$ when G is an affine algebraic unipotent group over $C$ with $\dim \left(G\right)=n$ and J is a Hopf 2-cocycle for G. The cotriangular Hopf algebras ${}_{J}O{\left(G\right)}_J$ have the same coalgebra structure as $O\left(G\right)$ 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 ${}_{J}O{\left(G\right)}_J$ is a crossed product $S{\#}_{\sigma }U\left(t\right)$, where $t$ is the Lie algebra of T and S is a deformed coideal subalgebra. Each simple ${}_{J}O{\left(G\right)}_J$-module factors through a unique quotient algebra ${}_{J}O{\left(Z_g\right)}_J$. These quotient algebras are parametrised by the double cosets TgT of T in G, and form an obvious direction for further study. The finite dimensional simple ${}_{J}O{\left(G\right)}_J$-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.