Twisting of affine algebraic groups, III

Abstract

Let $G$ be an affine algebraic group over $\C$ and let $J$ be a Hopf $2$-cocycle for $\Os(G)$. We prove that one-sided cocycle twisting preserves Gelfand--Kirillov dimension and that $\Os(G)_J$ is simple if and only if $J$ is minimal. If $H$ is the support of $J$, we identify the primitive and maximal spectra of $\Os(G)_J$ with the closed points of $G/H$ and show that all primitive quotients are isomorphic to the minimal support algebra $\Os(H)_J$. In particular, the prime and Jacobson radicals vanish, and finite-dimensional irreducible modules exist exactly when $H$ is finite. For connected solvable $G$, we construct an iterated differential Ore presentation over an algebraic quantum torus and deduce that $\Os(G)_J$ is an affine Noetherian domain. We describe the torus-trivial Hopf $2$-cocycles on $\Os(G)$ using nilpotent solutions of the classical Yang--Baxter equation and identify their supports with the algebraic hulls of their carriers. We also separate the centrally supported toral sector and formulate a relative Hopf $2$-cocycle problem for the remaining classes. Finally, we compute multi-extended Jordanian examples, study a McConnell-type family of quantum differential algebras, and establish a conditional birational deletion criterion and a normal form when the unipotent radical has dimension at most one.