**2-cocycle twists of Hopf algebras**

Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map $$ f: H \otimes H \to k$$ such that $$ f(x_{(1)},y_{(1)})f(x_{(2)} y_{(2)}, z) = f(y_{(1)}, z_{(1)}) f(x, y_{(2)} z_{(2)}) $$ (in Sweedler notation) and $$ f(x,1) = \varepsilon(x) = f(1,x) $$ for $x,y,z \in H$. Also one usually wants that $f$ is invertible for the convolution product, i.e. that there is some functional $$ \bar{f} : H \otimes H \to k $$ such that $$ f(x_{(1)}, y_{(1)}) \bar{f}(x_{(2)}, y_{(2)}) = \varepsilon (x) \varepsilon (y) = \bar{f} (x_{(1)}, y_{(1)}) f(x_{(2)}, y_{(2)}). $$

The cocycle can be used to twist the multiplication of $H$ as follows: $$ x \cdot_f y = f(x_{(1)}, y_{(1)}) x_{(2)} y_{(2)} \bar{f}(x_{(3)}, y_{(3)}), $$ The cocycle condition ensures that $\cdot_f$ is associative, and the fact that $f$ is unital means that the old identity element is still the identity element for $\cdot_f$.

You can also twist the antipode of the Hopf algebra in such a way as to get a new Hopf algebra structure on $H$, where the comultiplication and counit are the same as before. You have to check that the original comultiplication and counit are still algebra homomorphisms with respect to $\cdot_f$.

**Twisting the algebra structure only**

You can modify this construction in such a way as to obtain only an algebra instead of a Hopf algebra. This is done by defining $$ x \cdot_f y = f(x_{(1)}, y_{(1)}) x_{(2)} y_{(2)}. $$ Again, $f$ being unital means that the identity element is the same.

**The braided setting**

All of this can be done in exactly the same way whenever $H$ is a Hopf algebra object in a braided monoidal category. All the maps can be interpreted as string diagrams, and the proofs that $\cdot_f$ is associative and that the comultiplication and counit are algebra maps for the twisted multiplication go through in exactly the same way by manipulating the string diagrams.

**My question is:**

Does anybody know a reference for 2-cocycle twists of a Hopf algebra object in a braided monoidal category? I've looked in Majid's book, and he does talk a great deal about Hopf algebras in braided categories (what he calls braided groups), and also about 2-cocycle deformations, but I don't think he puts the two concepts together. Please correct me if I'm wrong. Thanks!