Jacobson Density Theorem

For an algebra $A$ over an algebraic closed field, there is

ii) Let $V=V_1\oplus \cdots \oplus V_n$, where $V_i$ are irreducible finite-dimensional pairwise non-isomorphic representations of $A$. Then $\rho =\oplus {\rho }_i:A\to \oplus \mathrm{End}(V_i)$ is surjective as well.

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When a $F$-algebra $A$ with $\overline{F}\ne F$, we have the following theorem.

Let $V$ be a irreducible finite-dimensional representation of $A$, and let $D={\mathrm{End}}_A(V)$. By Schur lemma, $D$ is a division ring and $V$ can be seen as a linear vector space over $D$, and one can check

$${\dim }_F(V)={\dim }_D(V)\cdot {\dim }_F(D).$$

Let $k={\dim }_D(V)$. The Jacobson density theorem tells us that $\rho (A)\simeq {\mathrm{M}}_k(D^{\mathrm{op}})$.

In particular, $V$ is an absolutely irreducible representation iff $\rho (A)={\mathrm{End}}_F(V)$ iff $D=F$.