AIMS Lie Groups: Fall-2022
HW 2 : Due 11 Mon 19 Dec

1. Generators of $SL(2,\mathbb{R})$ S0 4492S An explicit basis for the Lie algebra \(\mathfrak{sl}(2,\mathbb{C}\)) is given in GELG.
   1. Find a subbasis that spans the Lie subalgebra \(\mathfrak{sl}(2,\mathbb{R})\) of \(\mathfrak{sl}(2,\mathbb{C})\).
   2. Show that the commutators of these elements are linear combinations of these elements, thus verifying that the vector space \(\mathfrak{sl}(2,\mathbb{R})\) is in fact a Lie algebra.
   3. How many boosts (hyperbolic rotations) and how many (ordinary) rotations are in \(\mathfrak{sl}(2,\mathbb{R})\)? Explain your reasoning!
   4. The Killing form \(B\) on a (real) matrix Lie algebra \(\mathfrak{g}\) is given for \(X,Y\in\mathfrak{g}\) by \[B(X,Y)=\mathrm{tr}(XY)\] Determine the Killing form on \(\mathfrak{sl}(2,\mathbb{R})\). That is, determine the matrix representation of the Killing form when acting on the basis you found above.
2. Spin-2 Representation S0 4492S 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?