Download LIE ALGEBRAS QUESTION 1 Let k be an algebraically closed field

LIE ALGEBRAS
QUESTION 1
Let k be an algebraically closed field.
1.1. Let L be a 3-dimensional vector space over k with basis x, y, z. Give L an
anti-commutative algebra structure by setting
[x, y] = z, [y, z] = x, [z, x] = y,
and extending to L by k-linearity.
(a) Show that L is a Lie algebra.
(b) Compute the centraliser cL (x − 2y).
(c) Prove that L is a simple Lie algebra.
1.2. Verify that the following relations together with k-bilinearity and anticommutativity define a Lie algebra structure on L = span(a, b, c):
[x, y] = z, [x, z] = y, [y, z] = 0.
Compute the centre z(L) and the derived subalgebra L(1) . Show that L is solvable
but not nilpotent.
1.3. Let {e, h, f } be the standard basis of the Lie algebra sl(2, k). Compute the
matrices of the linear operators ad e, ad h and adf relative to this basis. Compute
[e, f + 3h] and [e − h + f, e − 2f ] and prove that (ad e)3 = 0.
1.4. Let A be an algebra over k (not necessarily associative or Lie) with multiplication (x, y) 7→ x · y. A linear operator D on the vector space A is called a
derivation of A if
D(x · y) = (Dx) · y + x · (Dy)
(∀ x, y ∈ A).
Verify that the commutator [D, D0 ] = D ◦ D0 = D0 · D of any two derivations of
A is again a derivation of A whereas the composition D ◦ D0 need not be.
1.5. Prove that a Lie algebra g is associative if and only if the derived subalgebra
of g is contained in the centre of g, that is g(1) ⊆ z(g).
1.6. Let g be a Lie algebra such that [[x, y], y] = 0 for all x, y ∈ g. Show that
3[[x, y], z] = 0 for all x, y, z ∈ g. [Hint: Observe that the mapping (x, y, z) 7→
[[x, y], z] is skew-symmetric in x, y, z and make use of the Jacobi identity.]