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.