T2: Basics

Question 1

Define the group U(1)

Answer 1

U(1) = {e^it; t ∈ R}

Question 2

Let G be a Lie group and g be its Lie algebra. Define the adjoint rep on G

Answer 2

The map from G to GL(g):
Adg(Y ) := gYg^-1

I.e. Takes an element in the Lie algebra and acts by conjugation.

Question 3

Define the adjoint group on g

Answer 3

The map g to ad_x in the End(g) = gl(g)

ad = D Ad

ad_x(Y) = [X,Y] = XY − YX.

Question 4

Define Ad_exp(tx)

Answer 4

= exp(t ad_x)

Question 5

Give the standard representation of a linear Lie group

Answer 5

ρ(g) = g
Dρ(X) = X