MATH 7364 Lecture 4 problems

# MATH 7364 Lecture 4 problems - PROBLEMS ON SYMPLECTIC...

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PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS 3. McKay correspondence upgraded (from last time) Exercise 3.3. A map C 2 C Γ C Γ extends to a representation from Rep Γ ( C x, y , C Γ) if and only if it is Γ -equivariant. Exercise 3.4. Show that Hom Γ ( C 2 C Γ , C Γ) = r i,j =0 M ij Hom C ( N i , N j ) = r i,j =0 Hom C ( N i , N j ) m ij = r i,j =0 Hom C ( C δ i , C δ j ) m ij . Note that the first equality is canonical, the second depends on the choice of a basis in M ij , while the third depends on the choice of bases in N i . 4. Deformed preprojective algebras Exercise 4.1. Show that C Q is associative and i Q 0 ϵ i is a unit in C Q . Further, show that, as a unital associative algebra, C Q is generated by ϵ i , i Q 0 , and a Q 1 subject to the relations ϵ i ϵ j = δ ij ϵ i , i Q 0 ϵ i = 1 , ϵ i a = δ ih ( a ) a, aϵ i = δ it ( a ) a . Exercise 4.2. Use the universal properties of all algebras involved to show that C x, y = T C Γ ( C 2 C Γ) and C Q = T ( C Q ) 0 ( C Q ) 1 .
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• Fall '12
• IvanLosev
• Math, Algebra, Functor, Mij, Ae ⊗eAe eM, Ni∗

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