Twists of quantum Borel algebras

TL;DR

This paper classifies all Drinfeld twists for quantum Borel algebras at roots of unity, revealing that they are generated by alternating forms on the character group, leading to a clear classification of related Hopf algebras.

Contribution

It provides a complete classification of Drinfeld twists for quantum Borel subalgebras at roots of unity using alternating forms, connecting twists to algebraic group actions.

Findings

All twists are generated by alternating forms on the character group.

Classification of Hopf algebras tensor equivalent to u_q(b).

Cocycle twists correspond bijectively to alternating forms.

Abstract

We classify Drinfeld twists for the quantum Borel subalgebra u_q(b) in the Frobenius-Lusztig kernel u_q(g), where g is a simple Lie algebra over C and q an odd root of unity. More specifically, we show that alternating forms on the character group of the group of grouplikes for u_q(b) generate all twists for u_q(b), under a certain algebraic group action. This implies a simple classification of Hopf algebras whose categories of representations are tensor equivalent to that of u_q(b). We also show that cocycle twists for the corresponding De Concini-Kac algebra are in bijection with alternating forms on the aforementioned character group.