T2: SL_2

Question 1

Define the conventional basis for sl_2 (C)

Answer 1

H = Id w/negative bottom right
X = 1 top right
Y = 1 bottom left

Question 2

State the commutator rules for the basis of sl_2 (C)

Answer 2

[H, X] = 2X
[H, Y] = -2Y
[X, Y] = H

Question 3

How can we decompose an arbitrary representation (ρ, V) of sl_2(C)?

Answer 3

V = ⨁V_α where each V_α is a complex eigenspace for H st. H(v) = αv

Question 4

Define a weight (in sl_2(C))

Answer 4

The eigenvalue corresponding to H(v) = αv

Question 5

Define a weight space and weight vector (in sl_2(C))

Answer 5

Each component of V (V_α) is a weight space and any vector in V_α is a weight vector

Question 6

Consider some vector v in V_α. Where do the actions of H,X,Y send v?

Answer 6

H(v) to V
X(v) to V_α+2
Y(v) to V_α-2

Question 7

What characterises the highest weight vector and highest weight?

Answer 7

The vector st
X(v) = 0, with v in V_α

α is the highest weight

Question 8

What characterises the lowest weight vector and lowest weight?

Answer 8

The vector st
Y(v) = 0, with v in V_α

α is the lowest weight

Question 9

How are the weights of sl_2 characterised?

Answer 9

A string of integers which differ by two and are symmetric about zero between -n and n.

Question 10

What are the only eigenvectors of X and Y?

Answer 10

The highest and lowest weight vectors respectively, with eigenvalue 0.

Question 11

What is the only finite dim unitary rep of SL_2(C)

Answer 11

The trivial rep

Question 12

How do weight vectors v_i and w_j combine for V⊗W given reps V and W

Answer 12

v_i ⊗ w_j ∈ V ⊗ W

is a weight vector in V ⊗ W with weight i+j

Question 13

Construct the weight vectors and weights of the nth power of the symmetric square Sym^n (C^2)

Answer 13

weight vectors: e1^k e2^n-k
weights: -n to n with 2k − n for each wv

Question 14

Given a highest weight vector v what subspace of V can we define?

Answer 14

Subspace spanned by vectors v, Y(v), …

Question 15

What is the highest (and lowest) weight for an irr rep sl2?

Answer 15

dimV = n+1

Question 16

How does H act on the tensor of two vectors?

Answer 16

Like a product rule

Question 17

How do we define a complexification?

Answer 17

V ⊗_R C

c(v ⊗ w) = v ⊗ cw for c in C

Question 18

Define the complexification of sl_n(R). What is this isomorphic to?

Answer 18

sl_n(R)_C = sl_n(R) ⊕ isl_n(R)