Twisting of affine algebraic groups, I

TL;DR

This paper studies the twisting of affine algebraic groups via Hopf 2-cocycles, analyzing the structure of associated twisted function algebras, especially for connected nilpotent groups, and classifying the resulting simple algebras.

Contribution

It introduces the support of the cocycle on a subgroup, describes the structure of twisted algebras for nilpotent groups, and classifies Hopf 2-cocycles for such groups.

Findings

J is supported on a conjugacy class of a subgroup H

O(G)_J is finitely generated with center O(G/H)

O(G)_J is a simple algebra if J is supported on G

Abstract

We continue the study of twisting of affine algebraic groups G (i.e., of Hopf 2-cocycles J for the function algebra O(G)), which was started in [EG1,EG2], and initiate the study of the associated one-sided twisted function algebras O(G)_J. We first show that J is supported on a closed subgroup H of G (defined up to conjugation), and that O(G)_J is finitely generated with center O(G/H). We then use it to study the structure of O(G)_J for connected nilpotent G. We show that in this case O(G)_J is a Noetherian domain, which is a simple algebra if and only if J is supported on G, and describe the simple algebras that arise in this way. We also use [EG2] to obtain a classification of Hopf 2-cocycles for connected nilpotent G, hence of fiber functors Rep(G)\to Vect. Along the way we provide many examples, and at the end formulate several ring-theoretical questions about the structure of the algebras O(G)_J for arbitrary G.