MATH7133 Semester 2, 2022 Assignment 3

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH7133 Semester 2, 2022

Assignment 3

1. (8 marks) The Lie algebra su(2) is generated by (e, f, h} subject to the commutation relations

[h, e] = 2e,

[h, f] = -2f,

[e, f] = h.

Verify that C = h2 + ef + fe is an element in the centre of the universal enveloping

algebra U (sl(2)).

2. (8 marks) Let L be a Lie algebra with basis (x1 , ..., xn } and commutation relations [xi , xj ] = Cij(k)xk .                                                     (1)

Show that the elements

Xi  =       Cji(k)aj k

j,k

also satisfy the commutation relations (1) with the aij  the usual basis elements for gl(n) satisfying

[aij , ak l ] = δj(k)ai l  - δl(i)akj .

This result is known as Ado’s theorem.

3. Consider the o(n) algebra with generators (aj k  = -akj  : 1 < j, k < n} satisfying the commutation relations

[aj k , ap q ] = δk(p)aj q  - δq(j)ap k  - δj(p)ak q + δq(k)apj .

Recursively define the elements

(A1 )j k  = aj k ,

n


(Am )j k  =


ajp (Am − 1 )p k .


p=1

(i) (4 marks) Show that

[aj k , (Am )p q ] = δk(p)(Am )j q  - δq(j) (Am )p k  - δj(p)(Am )k q + δq(k)(Am )pj .

n

(ii) (3 marks) Define Cm  =       (Am )j j . Show that

j=1

[aj k , Cm] = 0,           1 < j, k < n.

(iii) (3 marks) Show that 2C3  = (n - 2)C2 .


2

4. Consider the operators

1      1   d

2     2πi dx ,

e2πix          cos(2πx)  d 

2            2πi     dx ,

e2πix          sin(2πx)  d 

2i           2πi     dx

(i) (8 marks) Show that (Z1 , Z2 , Z3 } is closde under the commutator.

(ii) (4 marks) Show that (Z1 , Z2 , Z3 } provide a realisation of the su(2) algebra as

defined in Q.1.

(iii) (4 marks) Compute the realisation of the second-order su(2) Casimir invariant

C as defined in Q.1.

(iv) (4 marks) Determine all highest-weight states and lowest-weight states with

respect to your solution of part (ii). Also specify the associated highest weights and lowest weights.

5. (6 marks) Use the dimension formula on page 25 of the notes to calculate the dimen- sion of the irreducible gl(n)-module with highest weight 2∈1 - ∈n .

6. Consider the polynomial algebra C[x1 , x2 , x3] with x1 , x2 , x3   three independent, real, variables. Recall that the differential operators

aij  = xi                                                                                         (2)

satisfy the gl(3) commutation relations.

(i) (3 marks) Use Ado’s Theorem to find expressions for the o(3) generators L1 , L2 , L3 in terms of (2).

(ii) (5 marks) Show that

L2  = x.(x.V + 2)V - x.xV2

where

x = (x1 , x2 , x2 ),

V = ╱  ,  , 、 ,

V2  =  ∂2   +  ∂2   +  ∂2 

You will find it helpful to use the result

εijk εilm  = δjl δkm - δjm δkl .


2022-10-07