Lie Groups and Lie Algebras: Winter-2023
HW 3 : Due Wednesday 2/1

1. Spin-2 Representation S0 4608S Determine the matrices for the spin-2 representation of \(\mathfrak{su}(2)\). Then find all of the eigenvalues and eigenstates of one of your matrices. How many matrices should there be? What size are they? How many distinct eigenvalues are there?
2. The Lie algebra $\mathfrak{sl}(3,\mathbb{R})$ S0 4608S The Lie algebra \(\mathfrak{sl}(3,\mathbb{R})\) consists of all \(3\times3\) real matrices whose trace is zero.
   1. Find an orthonormal basis of \(\mathfrak{sl}(3,\mathbb{R})\).
      Orthonormal means orthogonal with respect to the Killing form \(B(X,Y)=\mathrm{tr}(XY)\), and with constant normalization (not necessarily \(1\)).
   2. What is the dimension of this Lie algebra? How many boosts are there?
   3. Find two orthogonal elements of \(\mathfrak{sl}(3,\mathbb{R})\) that commute with each other.
   4. Find all simultaneous eigenvectors of these two elements in \(\mathfrak{sl}(3,\mathbb{R})\).
      Don't forget which vector space you are in; the eigenvector equation has the form \([Q,X]=\lambda X\).
   5. Find a basis of \(\mathfrak{sl}(3,\mathbb{R})\) consisting entirely of simultaneous eigenvectors of your commuting operators.
   6. Compute at least 3 independent commutators of these eigenvectors.
   7. Make a table of the pairs of eigenvalues associated with each basis element. Plot these points in \(\mathbb{R}^2\).
   8. What are the angles between these points (measured at the origin)?