Topics in Geometry: Lie Groups and Lie Algebras

Lie Groups and Lie Algebras: Winter-2023
HW 5 : Due Wednesday 2/15

Root diagram of $\mathfrak{g}_2$ S0 4612S
1. Draw the root diagram of $\mathfrak{g}_2$. Identify two simple roots, and express each root as a linear combination of the simple roots.
  Work it out for yourself if necessary; see also these exercises.
2. Using the root diagram alone, identify an $\mathfrak{su}(3)$ subalgebra of $\mathfrak{g}_2$.
  Justify your answer briefly in words.
3. Can you find a second copy of $\mathfrak{su}(3)$ in $\mathfrak{g}_2$?
  Again, use only the root diagram in your argument, and briefly justify your answer.