the algebra $(A, \ast_{\mu})$. By virtue of $\mu$ satisfying this new algebra is associative and has the same algebra identity element as $(A,\cdot)$ (proved in Proposition \[prop: preserveassociative\]). We will use the notation $A^{G,\mu}:=(A, \ast_{\mu})$ and call such an object a cocycle twist of the induced $G$-grading on $A$ using the normalised 2-cocycle $\mu$. For brevity we may also use the term *cocycle twist* of $A$, since the notation makes clear both the group acting and the normalised 2-cocycle.

Perhaps the simplest nontrivial example of a cocycle twist is that of a twisted group algebra.

\[ex: twistedgroupalgebra\] Consider the group algebra $AG := A \otimes_k kG$, with $A$ and $G$ subject to the assumptions at the beginning of §\[subsec: theoryoftwist\]. The multiplication in this algebra is given by $$\label{eq: groupalgebramult}
(a \otimes_k g)(b \otimes_k h) = ab \otimes_k gh,$$ for all $a,b \in A$ and $g,h \in G$.

The group algebra has an obvious $G$-graded structure given by $A_g=\{a \otimes_k g : \; a \in A\}$. Twisting this grading using a normalised 2-cocycle $\mu$ produces a *twisted group algebra* $AG_{\mu}$. The multiplication in this algebra is defined by $$(a \otimes_k g) \ast_{\mu} (b \otimes_k h) =\mu(g,h)(ab \otimes_k gh),$$ for all 