given in Definition \[defn: ggradedalgebra\]. For now it suffices to state that for all $g\in G$ the corresponding component of the induced $G$-grading on $A$ is defined by $$\label{eq: inducedGgrading}
A_g := \{a \in A:\; a^h = \chi_{g^{-1}}(h)a,\; \forall \; h \in G \}.$$

Given such a grading, one can use a normalised 2-cocycle to deform the multiplicative structure of $A$.

\[def: normalised2cocycle\] Consider a function $\mu: G \times G \rightarrow k^{\times}$ satisfying the following relations for all $g,h,l \in G$ and the group identity element $e$: $$\label{eq: cocycleidinto}
\mu(g,h)\mu(gh,l)=\mu(g,hl)\mu(h,l), \;\;\; \mu(e,g)=\mu(g,e)=1.$$ We say that $\mu$ is a *normalised 2-cocycle* of $G$ with values in $k^{\times}$.

Although normalised 2-cocycles can be interpreted in terms of cohomology — as we do in §\[subsec: cocycletwists\] — for now we will use them solely to define a new multiplication on $A$. We will write $(A,\cdot)$ to denote the $G$-graded algebra $A$ and the standard multiplication $\cdot : A \times A \rightarrow A$, usually written as juxtaposition of elements.

One can define a new multiplication on the underlying $G$-graded vector space structure of $A$ as follows: for homogeneous elements $a \in A_g$ and $b \in A_h$ define $a \ast_{\mu} b:=\mu(g,h)ab$. This gives rise to 