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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.]