T2: Lie Groups and Lie Algebra

Question 1

Define a linear Lie group

Answer 1

A closed subgroup of GL_n(K).

Question 2

Let G be a (linear) Lie group. Define the Lie algebra

Answer 2

g = {X ∈ gl_n(C); exp(tX) ∈ G}

Question 3

Give three properties of the Lie algebra g

Answer 3

• g is a real vector space inside gl_n(K)
• If X ∈ g and if b ∈ G, then bXb^−1 ∈ g
• for X,Y in g: [X,Y] = XY-YX

Question 4

Give the formal definition of Lie algebra over field F

Answer 4

A F-vector space together with a Lie bracket:
[ , ] = g X g to g

Question 5

What are the three properties of the Lie bracket?

Answer 5

Bilinear, antisymmetric and satisfies the Jacobi identity.

Question 6

Under what condition is a Lie algebra abelian?

Answer 6

[X,Y] = 0 for any X,Y in g

Question 7

Given a Lie algebra g, define a Lie subalgebra h

Answer 7

A subspace h ⊂ g that is closed under the Lie bracket

Question 8

Define a Lie algebra homomorphism

Answer 8

A map between two Lie algebras st:
ϕ([X, Y ]) = [ϕ(X), ϕ(Y )]

And
ϕ(XY-YX) = [ϕ(X)ϕ(Y) - ϕ(Y)ϕ(X)]

Question 9

Given a Lie group homomorphism, how can we find a Lie algebra homomorphism?

Answer 9

Differentiating the Lie group hom and setting t = 0

Question 10

ϕ(exp(tX)) = ??

Answer 10

exp(tφ(X))

Question 11

Draw the commutative diagram for Lie groups and algebras

Answer 11

check phone

Question 12

φ(gXg^−1) = ??

Answer 12

ϕ(g) φ(X) ϕ(g^−1)

Question 13

how can we show φ is a Lie group homomorphism?

Answer 13

• Show it is R-linear
• Show it preserves the bracket.

Question 14

What are three properties of the Lie bracket?

Answer 14

• Bilinear
• Anti-symmetric
• Satisfies Jacobi

Question 15

Define the center of the Lie algebra

Answer 15

The element of the Lie algebra Z for which the Lie bracket is zero with any other element.