precise nature of the action will be given at the time.

Since $G$ is a finite abelian group, we can write it as a product of cyclic groups. We will denote the cyclic group of order $n$ by $C_n$. The Klein-four group will be pivotal in all of our examples from Chapter \[chap: sklyanin\] onwards, and using this notation it can be denoted by $(C_2)^2$.

#### Gradings

One of the fundamental definitions required for the work in this thesis is the following.

\[defn: ggradedalgebra\] The algebra $A$ is said to be *$G$-graded* if it possesses a $k$-vector space decomposition $A=\bigoplus_{g \in G}A_g$ for which $1 \in A_e$ and $A_g \cdot A_h \subset A_{gh}$ for all $g, h \in G$.

By replacing $G$ with ${\ensuremath{\mathbb{N}}}$ in this definition one obtains the definition of an *[$\mathbb{N}$]{}-graded algebra*.

We will make the additional assumption that $A$ is connected graded for certain results in §\[sec: preservation\]. Moreover, all of our examples of cocycle twists (apart from §\[subsec: homenvelopalg\]) involve such algebras.

\[def: conngraded\] A $k$-algebra $A$ is said to be *connected graded* or *c.g.* if it has an ${\ensuremath{\mathbb{N}}}$-grading $A = \bigoplus_{n \in {\ensuremath{\mathbb{N}}}}A_n$ for which $A_0=k$ and $\text{dim}_k A_n < \infty$ for all 