每日定理7

Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(1.14)

Let $A$ be an $F$-algebra. Then every irreducible $A$-module is isomorphic to a factor module of $A^\circ$. If $A$ is semisimple, then every irreducible $A$-module is isomorphic to a submodule of $A^\circ$.

Pf:

• Choose $0\neq v\in V$, an irreducible $A$-module
• Define $\vartheta:A\rightarrow V$ by $\vartheta(x)=vx$ and $\vartheta\in Hom_A(A^\circ,V)$
• $A^\circ/{ker\vartheta}\cong V$

Isaacs, $\textit{Character Theory of Finite Groups}$, Theorem(1.15)

Let $A$ be a semisimple algebra and let $M$ be an irreducible $A$-module. Then

1. $M(A)$ is a minimal ideal of $A$
2. if $W$ is irreducible, then it is annihilated by $M(A)$ unless $W\cong M$
3. the map $x\mapsto x_M$ is one-to-one from $M(A)$ onto $A_M\subseteq End(M)$
4. $\mathcal{M}(A)$ is a finite set

Pf: 1.

• The map $\vartheta_x: y\mapsto xy$ satisfies $\vartheta_x\in E_A(A^\circ)$
• $M(A)$ is an $E_A(A^\circ)$-module.

     2.

• If $W\not\cong M$, then $W(A)\cap M(A)=0$
• $W(A)M(A)\subseteq W(A)\cap M(A)=0$
• $A^\circ$ has a submodule $W_0\cong W$ and $W_0M(A)=0$
• Since $W\cong W_0$, they have the same annihilator in $A$