Twisting of affine algebraic groups, II

TL;DR

This paper investigates the algebraic structure and representations of twisted cotriangular Hopf algebras associated with connected nilpotent algebraic groups, revealing their properties and classifying their finite-dimensional irreducible representations.

Contribution

It provides a detailed analysis of the algebra structure, Noetherian properties, and representation theory of twisted cotriangular Hopf algebras for nilpotent algebraic groups, including explicit classifications.

Findings

${}_J ext{O}(G)_J$ is an affine Noetherian domain with Gelfand-Kirillov dimension equal to $ ext{dim}(G)$

If $G$ is unipotent and $J$ is supported on $G$, then ${}_J ext{O}(G)_J$ is isomorphic to the universal enveloping algebra $U( ext{Lie}(G))$

Finite-dimensional irreducible representations are classified via twisted function algebras on double cosets

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.