PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS3.McKay correspondence upgraded (from last time)Exercise 3.3.A mapC2⊗CΓ→CΓextends to a representation fromRepΓ(C⟨x, y⟩#Γ,CΓ)if and only if it isΓ-equivariant.Exercise 3.4.Show thatHomΓ(C2⊗CΓ,CΓ) =r⊕i,j=0Mij⊗HomC(N∗i, N∗j)=r⊕i,j=0HomC(N∗i, N∗j)⊕mij=r⊕i,j=0HomC(Cδi,Cδj)mij.Note that the first equality is canonical, the second depends on the choice of a basis inMij,while the third depends on the choice of bases inN∗i.4.Deformed preprojective algebrasExercise 4.1.Show thatCQis associative and∑i∈Q0ϵiis a unit inCQ. Further, showthat, as a unital associative algebra,CQis generated byϵi, i∈Q0,anda∈Q1subject to therelationsϵiϵj=δijϵi,∑i∈Q0ϵi= 1, ϵia=δih(a)a, aϵi=δit(a)a.Exercise 4.2.Use the universal properties of all algebras involved to show thatC⟨x, y⟩#Γ∼=TCΓ(C2⊗CΓ)andCQ∼=T(CQ)0(CQ)1.
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