Drinfeld twists of Koszul algebras

Abstract

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 A μ . We show that when A is a quadratic algebra, and H acts on A by degree-preserving endomorphisms, then the twist A μ is also quadratic. Furthermore, if A is a Koszul algebra, then A μ 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 ( s l 2 ) is a quadratic algebra equal to the q − 1 -quantum plane.