Physics 129B, Winter 2010
Problem Set 1 Solution
Chan Y. Park
February 8, 2010
Problem 1
Show that any nite groups of orders 2 and 3 are isomorphic to the cyclic groups C2 and C3 .
(Hint : What are possible multiplication tables?)
Solution:
Consider a gro
Physics 129B, Winter 2010
Problem Set 3 Solution
Chan Y. Park
February 11, 2010
Problem 1
Suppose that, for any element g of a group G, we have a relation g 2 = e. Show that G is an Abelian
group, i.e. g1 g2 = g2 g1 for any pair of elements g1 , g2 G.
Sol
Physics 129B, Winter 2010
Problem Set 4 Solution
H.-J. Chung
February 19, 2010
Problem 1
[1-A] Consider the dihedral group D6 , which is the symmetry group of an equilateral triangle on
the xy plane. We assume that the center of the triangle coincides wit
Physics 129B, Winter 2010
Problem Set 5 Solution
Chan Y. Park
March 3, 2010
Problem 1
Four equal masses m are connected by six springs of spring constant k in such a way that the
equilibrium positions of the masses are at the corners of a regular tetrahed
Physics 129B, Winter 2010
Problem Set 6 Solution
H.-J. Chung
March 9, 2010
Problem 1
(a) The orthogonal group O(n) acts on the n-dimensional Euclidean space Rn . Show that the Lie
algebra of O(n) is generated by
Lab = i(eab eba ),
(a, b = 1, 2, , n; a < b
Physics 129B, Winter 2010
Problem Set 7 Solution
Chan Y. Park
March 9, 2010
Problem 1
Suppose , are root vectors and E , E are corresponding raising operators. Show that [E , E ]
must be proportional to E+ . What happens if ( + ) is not a root?
Solution:
Physics 129B, Winter 2010
Problem Set 4 Solution
H.-J. Chung
March 18, 2010
Problem 1
(a) Show that the SU (N ) Lie algebra has an SU (N 1) subalgebra.
(b) The SU (N ) Lie group consists of N N unitary matrices with unit determinant. Thus, it
has a natura
Physics 129B, Winter 2010
Problem Set 9 Solution
Chan Y. Park and H. -J. Chung
March 22, 2010
Problem 1
Suppose ui transforms as a 3 and v jk transforms as a 6 of SU (3) where i, j, k = 1, 2, 3. Decompose
the product of the tensor components ui v jk into