4.

There is an action of $G$ by $k$-algebra automorphisms on $M_2(k)$, the ring of $2 \times 2$ matrices over $k$. This action is defined by $$\label{eq: matrixaction}
M^{g_{1}} =\begin{pmatrix}
-1 & 0 \\
0 & 1 
\end{pmatrix}M\begin{pmatrix}
-1 & 0 \\
0 & 1 
\end{pmatrix},\; M^{g_{2}}=\begin{pmatrix}
0 & 1 \\
1 & 0 
\end{pmatrix}M\begin{pmatrix}
0 & 1 \\
1 & 0 
\end{pmatrix},$$ for a matrix $M \in M_2(k)$ and the group generators $g_1$ and $g_2$. A 2-cocycle arises here via an action of $G$ on $k^2$ – such objects are important for the constructions we will soon describe.

Now consider the tensor product of $k$-algebras $A \otimes_{k} M_2(k)$. Given the action of $G$ in and the embedding $G \rightarrow \text{Aut}_{\text{alg}}(A)$, there is a natural diagonal action of $G$ on the tensor product, where $$\label{eq: diagaction}
(a \otimes_k M)^g = a^g \otimes_k M^g,$$ for all $a \in A$ and $M \in M_2(k)$.

Odesskii’s construction concludes by taking the invariant ring of $A \otimes_{k} M_2(k)$ under this diagonal action. We highlight this algebra for future reference.

\[ex: odesskii\] Odesskii’s example is given by the invariant ring $(A \otimes_{k} M_2(k))^G$.

It is natural to ask if the properties of $A$ are shared 