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Math 662 Quiver Representations Homework Assignment 3 Due Friday March 7 Let k be an algebraically closed field. All modules are right modules. 1. Let Q be the quiver α •R β Let n be a positive integer, and let M = k n be the representation of Q for which 0 1 0 ··· 0 0 .. 1 0 0 1 . ... . . 0 1 φα = . . 0 and φβ = . . . .. .. .. 1 0 ··· 0 1 0 0 Show that M is simple. 2. Let Q be the quiver α 1 * 2 • •j β Show that kQ has infinitely many simple modules. 3. Let Q be the quiver 2 β 4 •` •_ α •1 δ 3• γ ~ Describe all simple kQ-modules and all projective indecomposable kQmodules (by giving a nice vector space basis of each). 4. Let Q be the Kronecker quiver α 1 • 5 * 2 • β For each λ ∈ k, let Hλ be the representation of Q for which (Hλ )1 = k, (Hλ )2 = k, φα = id, and φβ = λid, where id is the identity map. Show that, for each λ, Hλ is indecomposable, and that Hλ ∼ = Hµ if and only if λ = µ.