Questions: Root Systems of Semisimple Lie Algebras

Question 1

How many roots does sl₃(ℂ) have, and what is its type?

• A. 2 roots, type A₁
• B. 6 roots, type A₂
• C. 8 roots, type A₂
• D. 6 roots, type G₂

Answer: (B)

sl₃(ℂ) has dimension 8. Its Cartan subalgebra of traceless diagonal matrices has dimension 2, leaving 6 one-dimensional root spaces, one for each root eᵢ - eⱼ with i ≠ j. They form the hexagon of type A₂. G₂ has 12 roots.

Question 2

If α and β are roots of a reduced root system, then 2(β, α)/(α, α) is an integer.

• T. True
• F. False

Answer: True

This is the integrality axiom. For a Lie algebra it comes from the sl₂-subalgebra attached to α: β evaluated on its coroot is a weight of a finite-dimensional sl₂-representation, and such weights are integers.

Question 3

What group is generated by the reflections in the roots of sl₃(ℂ), and what is its order?

Model answer: The Weyl group of type A₂, isomorphic to the symmetric group S₃, of order 6.

The reflection in eᵢ - eⱼ swaps the i-th and j-th coordinates, so the reflections generate all permutations of three coordinates. In general the Weyl group of type A_n is S_{n+1}.

Question 4

Why must the angle between two non-proportional roots be one of finitely many values?

Model answer: The product of the integers 2(β, α)/(α, α) and 2(α, β)/(β, β) equals 4cos²θ, which must be an integer between 0 and 3, so cos²θ is 0, 1/4, 1/2 or 3/4.

Those values give angles of 90°, 60° or 120°, 45° or 135°, and 30° or 150°. The value 4 is excluded because it would make the roots proportional. This finiteness is the starting point of the classification.