MATH 7364 Lecture 7 problems

MATH 7364 Lecture 7 problems - PROBLEMS ON SYMPLECTIC...

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PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS 7. Hochschild cohomology and deformations Exercise 7.1. Let A 0 be an algebra and A 1 = A 0 P A 0 its first order deformation with product defined by μ ( a, b ) = ab + μ 1 ( a, b ) for a, b A 0 , where μ 1 P C 2 ( A, A ) . Show that the product μ is associative iff 1 = 0 . Exercise 7.2. Let A 1 , A 1 = A 0 P A 0 be two 1st order deformations of A 0 with products μ, μ . Let σ : A 1 A 1 be an S ( P ) -module isomorphism such that σ ( a ) = a + σ 1 ( a ) for a A 0 , where σ 1 P C 1 ( A, A ) . Show that σ is an algebra homomorphism iff μ = μ + . Exercise 7.3. Let A k be a deformation of A 0 over S ( P ) / ( P k +1 ) , A k = k i =0 S i ( P ) A 0 . Let μ ( a, b ) = ab + k i =1 μ k ( a, b ) be the product. Consider the equation (1) μ k ( a, b ) c + μ k ( ab, c ) μ k ( a, bc ) k ( b, c ) = k 1 i =1 ( μ i ( a, μ k i ( b, c )) μ i ( μ k i ( a, b ) , c )) . Show that the r.h.s. is a cocycle if k = 2 . Problem 7.1. Show that the r.h.s. of (1) is a cocycle for any k .
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