How to Find the automorphism group of the $A$-module $V$

Question

Let $A$ be a semi-simple finite dimensional algebra over complex filed $C$, and let $V$ be a direct sum of two isomorphic simple $A$-modules. Find the automorphism group of the $A$-module $V$.

My thoughts: based on the definition of semi-simple finite algebra, we can know that $A$ can be represented by $A\cong A_1\oplus A_2$, where $A_i$ is simple finite algebra. Then, we feel a little confused how to analyze this question, can someone help me?

Here is the definition of semisimple module:https://en.wikipedia.org/wiki/Semisimple_module

Answer 1

Fix a decomposition $V=W\oplus W$ where $W$ is a simple $A$-module. $Aut_{A}(V,V)=GL_2(End_{A}(W,W))$. By Schur's lemma, $End_{A}(W,W)\cong C$. Thus $Aut_A(V,V)=GL_2(C)$.