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.

pdf/drinfeld-twists-of-koszul-algebras-2c2rqq9n.pdf

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. 

The main result of the paper is that braided Cherednik algebras introduced by the first two authors are cocycle twists of rational Cherednik algebras. 
This gives a new construction of mystic reflection groups and a new proof that such groups have Artin-Schelter regular rings of quantum polynomial invariants.
Furthermore, the main result leads to a construction of 
finite-dimensional representations of braided Cherednik 
algebras. 

In this first version of the paper, we give a full proof of the main result and sketch the application to representations of braided Cherednik algebras.

Twists of rational Cherednik algebras

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Twists of rational Cherednik algebras