Lie-Groups und Representation Theory Exercise Sheet 12 Summer Term 2019 Prof. Dr. Johannes Walcher, Sebastian Nill, Lukas Hahn

Deadline: 09.07.2019, 11:00 Uhr

Exercise 12.1 Semisimplicity Let g be a Lie algebra. Show that the following statements are equivalent. (i) g is semisimple. (ii) g has no non-trivial abelian ideals. (iii) g has no non-trivial nilpotent ideals. (iv) g has no non-trivial solvable ideals. (v) The nilradical is zero, nil(g) = 0. (vi) The radical is zero, rad(g) = 0. (vii) The Killing form of g is non-degenerate. (viii) g is a direct sum of simple ideals. You may employ all results from the lecture.

Exercise 12.2 On the Killing Form Let g be a Lie algebra. (a) Suppose a is an ideal of g. Show that the restriction of the Killing form of g to a is the Killing form of a, i.e. Ba = Bg|a×a. (b) Show that if g is real and its Killing form B is positive definite, g must be trivial. Hint: Firstly show that ad-invariance of the Killing form implies aijk = −aikj for P the structure constants in [xi, xj] = k aijkxk, where (xi) is an ONB of (g,B). (c) Give an example of a semisimple real Lie algebra whose Killing form is not negative definite.

(d) Bonus: Any complex semisimple Lie algebra g has a compact real form, i.e. there is a

real Lie algebra gR such that its complexification is g = gR ⊗R C and its Killing form B is negative definite. Show this statement in the case of classical Lie algebras. gR

Exercise 12.3 A very useful trace formula Show that for g = sl(n, C) the Killing form is given by Bg(x, y) = 2n · tr(xy). Use this result to show that sl(n, C) is simple. Exercise 12.4 Accidental isomorphisms We will show the isomorphisms

SU(4) =∼ Spin(6) and Sp(2) =∼ Spin(5) where Spin(n) denotes the universal cover of SO(n) and Sp(n) = Sp(2n, C) ∩ U(2n) ⊂ SU(2n) is the compact symplectic group, which is simply connected and indeed compact. The symplectic group is defined by    2n×2n T 0n 1n 2n×2n Sp(2n, C) = X ∈ C | X JX = J where J = ∈ C . −1n 0n

1 Lie-Groups and Representation Theory - Summer Term 2019

Throughout this exercise, you may assume that all appearing Lie algebras are simple.

(a) Let V be a 4-dimensional complex vector space, W = Λ2V ∗ be the space of anti- symmetric bilinear forms on V and U = Λ4V ∗ be the space of volume forms on V . Denote the wedge product of two-forms by Q : W × W → U. By choosing a volume ∼ form, we get an identification U = C. Thus, Q becomes a bilinear form on W . Show that Q is symmetric and non-degenerate.

(b) Let GQ be the subgroup of GL(W ) that leaves Q invariant. Identify W as a repre- sentation of SL(4, C) induced from the standard representation on V . Show that Q is invariant under SL(4, C) and thus there is a homomorphism SL(4, C) → GQ. (c) Show that there is an isomorphism SU(4) → Spin(6). Hint: You may use the fact that any compact subgroup of

 n×n T SO(n, C) = X ∈ C | X X = 1n, det(X) = 1

is contained in an SO(n).

(d) Show that this isomorphism restricts to an isomorphism Sp(2) → Spin(5).

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