Drinfeld twists of Koszul algebras

Abstract

Given a Hopf algebra $H$ and a counital $2$-cocycle $\mu$ on $H$, Drinfeld introduced a notion of twist which deforms an $H$-module algebra $A$ into a new algebra $A_\mu$. We show that when $A$ is a quadratic algebra, and $H$ acts on $A$ by degree-preserving endomorphisms, then the twist $A_\mu$ is also quadratic. Furthermore, if $A$ is a Koszul algebra, then $A_\mu$ is a Koszul algebra. As an application, we prove that the twist of the $q$-quantum plane by the quasitriangular structure of the quantum enveloping algebra $U_q(\mathfrak{sl}_2)$ is a quadratic algebra equal to the $q^{-1}$-quantum plane.