an Example about Complexification

Consider a group representation

$$\rho :{\mathbb{Z}}_4\to \mathrm{GL}(2,\mathbb{R}),a\mapsto \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),$$

and note that it is a irreducible $\mathbb{R}$-module and is a reducible $\mathbb{C}$-module. Also note that

$$\rho (\mathbb{R}{\mathbb{Z}}_4)=\left\{\left(\begin{array}{cc}x& y\\ -y& x\end{array}\right):x,y\in \mathbb{R}\right\}:=D,$$

which deduces that $\rho (\mathbb{R}{\mathbb{Z}}_4)\ne {\mathrm{M}}_2(\mathbb{R})$, coincides with Jacobson Density Theorem.

Consider ${\mathrm{End}}_{{\mathbb{Z}}_4}({\mathbb{R}}^2)$, and one can check that it equals

$$C_{\mathrm{GL}(2,\mathbb{R})}(\rho (a))=D.$$

Since $D\simeq \mathbb{C}$ by $\left(\begin{array}{cc}x& y\\ -y& x\end{array}\right)\mapsto x+yi$, there is $\mathrm{Aut}(D)=⟨\sigma ⟩\simeq {\mathbb{Z}}_2$ generated by an automorphism induced by conjugation.

Identify ${\mathbb{R}}^2$ as a $D$-module, denoted by ${\mathbb{R}}_1^2$. Identify ${\mathbb{R}}^2$ as a $\sigma (D)$-module, that is, $d\ast v=\sigma (d)v$, denoted by ${\mathbb{R}}_2^2$. Then we can verify that

$${\mathbb{R}}_1^2\oplus {\mathbb{R}}_2^2\simeq \mathbb{C}\otimes {\mathbb{R}}^2,\text{\hspace{1em}(*)}$$

where $\mathbb{C}\otimes {\mathbb{R}}^2$ is a $\mathbb{C}\otimes \mathbb{R}{\mathbb{Z}}_4$-module.

To verify $(\ast )$, note that action on ${\mathbb{R}}_1^2$ can be written as

$$\left(\begin{array}{cc}x& y\\ -y& x\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}ax+by\\ -ay+bx\end{array}\right)$$

which corresponds to

$$\overline{(x+yi)}(a+bi)=(ax+by)+i(bx-ay).$$

For any $\left(\begin{array}{c}a\\ b\end{array}\right)\oplus \left(\begin{array}{c}c\\ d\end{array}\right)\in {\mathbb{R}}_1^2\oplus {\mathbb{R}}_2^2$ and $\left(\begin{array}{cc}x& y\\ -y& x\end{array}\right)\oplus \left(\begin{array}{cc}p& q\\ -p& q\end{array}\right)\in \overline{D}\oplus D$, the action corresponds to

$$\left(\left(\begin{array}{cc}x& y\\ -y& x\end{array}\right)\oplus \left(\begin{array}{cc}p& q\\ -q& p\end{array}\right)\right)\left(\left(\begin{array}{c}a\\ b\end{array}\right)\oplus \left(\begin{array}{c}c\\ d\end{array}\right)\right)=((x+yi)\otimes (p-qi))\cdot \left((a+bi)\otimes (c+di)\right).$$

Remark. In Jacobson Density Theorem, we have ${\dim }_F(V)={\dim }_D(V)\cdot {\dim }_F(D)$. For this case, ${\dim }_F(V)={\dim }_{\mathbb{R}}({\mathbb{R}}^2)=2$, ${\dim }_D(V)={\dim }_D({\mathbb{R}}^2)=1$ and ${\dim }_F(D)={\dim }_{\mathbb{R}}(D)=2$. Here ${\dim }_{\mathbb{R}}(D)=|\mathrm{Aut}(D)|$, which not always hold, see here.