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Problem set 4. Due Wednesday, October 28. Mathematics 534, Term 1, 2015. Instructor: Reichstein. All Lie algebras and all vector spaces (in particular, all reprsentations of Lia algebras) in the problems below are assumed to be finite-dimensional and defined over the field C of the complex numbers. 1. Let B : V ×V → V be a non-degenerate symmetric bilinear form on an n-dimensional vector space V . Recall that the orthogonal complement W ⊥ of a subspace W ⊂ V , relative to B, is defined as W ⊥ = {v ∈ V | b(v, w) = 0 ∀w ∈ W } . (a) Show that dim(W ) + dim(W ⊥ ) = dim(V ). (b) Show that (W ⊥ )⊥ = W . 2. Let L be a Lie algebra and B : L × L → C be a non-degenerate symmetric bilinear form. Suppose B is associative, i.e., B([x, y], z) = B(x, [y, z]) for every x, y, z ∈ L. Show that y ∈ L lies in the image of ad(x) : L → L if and only if B(y, z) = 0 for every z ∈ L, which commutes with x. Hint: Use Problem 1 with V = L and W = image of ad(x). 3. Let V and W be L-modules. In class, we defined the structure of an L-module on Hom(V, W ) via (a · f )(v) := a · (f (v)) − f (a · w) for any a ∈ L, f ∈ Hom(V, W ), and v ∈ V . In particular, in the special case, where W = C, with trivial L-action, V ∗ := Hom(V, C) is an L-module via (a · f )(v) := −f (a · w). for any a ∈ L, f ∈ V ∗ , and v ∈ V . We have also defined an isomorphism between the tensor product V ⊗W and Hom(W ∗ , V ) which identifies v ⊗ w ∈ V ⊗ W with the element of Hom(W ∗ , V ), taking l ∈ W ∗ to l(w)v. This endows V ⊗W with the structure of an L-module. Show that this L-module structure is given by a · (v ⊗ w) 7→ (a · v) ⊗ w + v ⊗ (a · w) . 4. Pages 30-31, Problem 5(a), (d). 5. Page 31, Problem 6. 6. Page 31, Problem 7. Note that we have shown that sln is simple; this was one of the problems on Assignment 1.