v ∈ U. Here aγ ∈ Hom(Vγ , Vαγ ). Denote aγk by ak, k = 1, . . . , p. Note that if γ ∈ Γk, αγ ∈ Γm then emaek ∈ Aα, so P aγ ∈ Aα for every k = 1, . . . , p. γ∈Γk If αγk ∈/ Γ0 then Wα,γ = {b ∈ Hom(Vγ , Vαγ ) | ∃a ∈ Aα : aγ = b} 6= 0 for every γ ∈ Γk. Since Wα,γ is a (End Vαγ )-(End Vγ )-bimodule embedded into Hom(Vγ , Vαγ ), we have Wα,γ = Hom(Vγ , Vαγ ). Given γ ∈ Γk, α ∈ Γ, αΓk = Γm, the map σα,γ : ak 7→ aγ, a ∈ Aα, is a linear bijection between Hom(Vγk , Vαγk ) and Hom(Vγ , Vαγ ). Assume σα,γk = id and σe,γ (x) = ιγ xιγ−1, since we have already established the structure of Ae. Suppose a = P akσα,γ ∈ Aα, b = P ιβbmιβ−1Tβ ∈ Ae, and consider γ∈Γk β∈Γm

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