Twisting of affine algebraic groups, II (2009.07760v2)

Abstract: We use \cite{G} to study the algebra structure of twisted cotriangular Hopf algebras ${}J\mathcal{O}(G){J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}J\mathcal{O}(G){J}$ is an affine Noetherian domain with Gelfand-Kirillov dimension $\dim(G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}J\mathcal{O}(G){J}\cong U(\g)$ as algebras, where $\g={\rm Lie}(G)$. We also determine the finite dimensional irreducible representations of ${}J\mathcal{O}(G){J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $H\subset G$ of $J$. Finally, we work out several examples to illustrate our results.