$ (\DeclareMathOperator{\End}{End} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\Imm}{Im} \DeclareMathOperator{\Hom}{Hom} $
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⟦𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧. A *homomorphism of algebras* ⁅f:A→B⁆ is a linear map such that ⁅f(x ⋅↙A y)=f(x) ⋅↙B f(y) &&∀x,y∈A⁆⟧ ⟦𝐓𝐡𝐞𝐨𝐫𝐞𝐦. Let ⁅V₁,V₂⁆ be representations of an algebra ⁅A⁆ over any field ⁅F⁆. Let ⁅ϕ:V₁→V₂⁆ be a nonzero homomorphism of representations. Then: 1. If ⁅V₁⁆ is irreducible then ⁅ϕ⁆ is injective. 2. If ⁅V₂⁆ is irreducible then ⁅ϕ⁆ is surjective. So, if both ⁅V₁,V₂⁆ are irr., then ⁅ϕ⁆ is an isomorphism.⟧ ⟦𝐏𝐫𝐨𝐨𝐟. 1. The kernel of ⁅ϕ⁆ is a subrepresentation of ⁅V₁⁆ which cannot be ⁅V₁⁆. So by irreducibility of ⁅V₁⁆ we have that the kernel is ⁅0⁆. 2. The image ⁅I⁆ of ⁅ϕ⁆ is a subrepresentation of ⁅V₂⁆ that cannot be ⁅0⁆. So by irreducibility of ⁅V₂⁆ we have ⁅I=V₂⁆. ⟧
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