Physics 467
Homework No. 10
1.
Solutions:
due Wednesday, 13 April, 2011
Let us consider the Lie algebras su(n + 1), the real, compact form of An , which have a simple
dening representation as the set of all traceless (n + 1) (n + 1)-matrices. A (standard)
Physics 467
Homework No. 5 Solutions: due Wednesday, 23 February, 2011
1.
Taking the identity 5 5 matrix as the metric in 5-dimensions, the generators for rotations in
that space must be skew-symmetric matrices. In 5-dimensional space there are
1
2 (5)(4)
Physics 467
Homework No. 4
1.
Solutions:
due Friday, 18 February, 2011
Please consider the rank 2 Lie algebra with the following Cartan matrix:
A=
+2
1
2
+2
.
Labeling the two basis vectorsin the Chevalley normalizationfor the Cartan subalgebra as
H1 and
Physics 467
Homework No. 3 Solutions:
1.
due Wednesday, 9 February, 2011
Please consider the following set of 8 matrices, which it is claimed constitute a (simple) Lie
algebra. The symbol below is simply a shorthand notation for a particular cube-root of
Physics 467
Homework No. 2 Solutions:
1.
due Wednesday, 2 February, 2011
Consider the following three dierential operators:
a1 2qq ,
a 2 q 2 q ,
a3 q ,
where the notation q means /q , i.e., the derivative with respect to the variable q , where of
course t
Physics 467
Homework No. 1
1.
Solutions:
due Friday, 28 January, 2011
The eight eighth-roots of unity form a nite, 8-dimensional group. Take them in the representation cfw_gm eim/4 | m = 1, 2, . . . , 8, with the product as ordinary multiplication of comp
Physics 467
Homework No. 11
1.
Solutions:
due Wednesday, 20 April, 2011
Use the method that creates representations of the loop group over a Lie algebra as innite
by innite matrices, create such representations of the following elements of (A2 ), where we
Physics 467
Homework No. 9
1.
Solutions:
due Wednesday, 6 April, 2011
Please consider the broken su(3) symmetry of the basic octet of fermionsthe one that includes the proton and the neutronand calculate a relation between the masses in the octet,
and the
Physics 467
Homework No. 8
1.
Solutions:
due Wednesday, 30 March, 2011
Determine all the partitions of 6, and display the associated Young diagrams. Use the hooklength formula to determine the dimension of the associated representations of S6 . How many
d
Physics 467
Homework No. 7
1.
Solutions:
due Wednesday, 9 March, 2011
As a rank-2 Lie algebra, any irreducible representation of G2 can be specied by giving two
integers, [m, n], the Dynkin coecients for the two reciprocal basis vectors in H , to describe
Physics 467
Homework No. 6
1.
Solutions:
due Wednesday, 2 March, 2011
Please use our discussions and/or Cahns help and show me all the details needed to determine
the various weights for the representation [2, 2] of su(3), which lives in a 27-dimensional