Twists of quantum Borel algebras

Abstract

We classify Drinfeld twists for the quantum Borel subalgebra uq(b) in the Frobenius–Lusztig kernel uq(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 uq(b) generate all twists for uq(b), under a certain algebraic group action. This implies a simple classification of finite-dimensional Hopf algebras whose categories of representations are tensor equivalent to that of uq(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.