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.
Related
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BibTeX
@misc{gelaki2026twisting,
title = {{Twisting of affine algebraic groups, III}},
author = {Shlomo Gelaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29948},
eprint = {2610.29948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29948},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twisting of affine algebraic groups, III
AU - Shlomo Gelaki
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29948
UR - https://arxiv.org/abs/2610.29948
AB - 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.
ER -