MATH 7364 Lecture 9 problems

# MATH 7364 Lecture 9 problems - PROBLEMS ON SYMPLECTIC...

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PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS 9. Commutativity and centers Exercise 9.1. Show that , ·} t,c = t , ·} , where , ·} is the standard bracket on S ( V ) Γ . Exercise 9.2. Prove the commutativity theorem in the case when V is not necessarily sym- plectically irreducible. Exercise 9.3. Let A be a Z > 0 -filtered algebra. If gr A is finitely generated, then so is A . Problem 9.1. Let p P 0 . Equip Z p with a filtration restricted from H p . Show that gr Z p = S ( V ) Γ . Deduce that H p is a finitely generated module over Z p . Problem 9.2. Now let p ̸∈ P 0 . Show that the center of H p coincides with C as follows: (1) Let z lie in the center of H p . Show that gr z gr H p = S ( V )#Γ actually lies in S ( V ) Γ . (2) Show that gr z lies in the Poisson center of S ( V ) Γ , meaning that { gr z, S ( V ) Γ } = 0 . (3) Show that the Poisson center of S ( V ) Γ coincides with C . Problem 9.3. In this problem we are going to equip Z c with a structure of a Poisson algebra. Fix c and consider H t,c as an algebra over C [ t ] by making t an independent variable. (1) Let a, b Z c . Lift a, b H c = H t,c / ( t ) to elements ˜ a, ˜ b
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Unformatted text preview: b ∈ H t,c . Show that [˜ a, ˜ b ] ∈ tH t,c and that the element 1 t [˜ a, ˜ b ] modulo t depends only on a, b and lies in Z c . Let { a, b } be that element. Show that {· , ·} is the Poisson bracket. (2) Show that { Z 6 i c , Z 6 j c } ⊂ Z i + j-2 c . Show that the induced bracket on gr Z c = S ( V ) Γ is a nonzero multiple of the standard bracket. Can you identify the scalar factor? Problem 9.4. Show that the scheme C p is irreducible and normal (and, well, Cohen-Macaulay and Gorenstein, if you know what these words mean). Problem 9.5. Show that if C p is smooth, then H p e is a locally free H p-module. Problem 9.6. Let A be a ﬁltered algebra. Show that if gr A has ﬁnite global dimension, then A does. Problem 9.7. Prove that if C p is smooth, then p is spherical (we deal here with p ∈ P ). 1...
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