Drinfeld twists of Koszul algebras

Given a Hopf algebra H and a counital 2-cocycle μ on H, Drinfeld introduced a notion of twist which deforms an H-module algebra A into a new algebra Aμ. We show that when A is a quadratic algebra, and H acts on A by degree-preserving endomorphisms, then the twist Aμ is also quadratic. Furthermore, if A is a Koszul algebra, then Aμ 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 Uq(sl2) is a quadratic algebra equal to the q−1-quantum plane.