T2: SL_3 Flashcards
Define Cartan sub-algebra h
A maximal abelian subalgebra whose adjoint action on g can be simultaneously diagonalized. There exists a basis Xi of g st:
[H, Xi] = α(H)Xi for all H in h
What is the standard Cartan subalgebra for sl_3?
3d Diagonal matrix with tr = 0
Is h abelian? Why?
Yes, because diagonal matrices commute with one another.
Define the matrix E_ij
The matrix with a single value one in row i column j.
Define the commutators of [E_ab, E_cd]
[E_ab, E_cd] = δ_bc E_ad − δ_ad E_cb
How do we define H_12 and H_23 (cartan)
H_12 = E_11 - E_22
H_23 = E_22 - E_33
How do we define the X and Y analogues in sl_3
X: upper triangular: E_12, 13, 23
Y: lower triangular: E_21, 31, 32
Write the commutator relations which define the weights of sl_3
[H, E_12] = α_1(H)E_12
[H, E_23] = α_2(H)E_23
[H, E_13] = α_3(H)E_13
Flip the coeffs of e, flip the sign
What relationship define α_1,2,3?
α_1 + α_2 = α_3
Define the simple roots (of sl_3)
{α_1, α_2}: the positive roots which cannot be written as the sum of two elements of Φ+
Give the root space decomp of Lie algebra g
g = h ⊕_α g_a
for cartan subalgebra h and root spaces g_α. (consider h as root space g_0 with 2d vs 1 for other roots)
Give an expression for the functional L_i
Diagonal matrix w/α_i α_2 α_3
How do we calculate the functional Li ?
Use the basis matrices (i.e H_12 and H_23)
I.e. L_1 is a the row vector with elements given by the top left components of H_12 and H_23
What are the relationships between α_i and L_i for the adjoint rep?
α1 = L1 − L2,
α2 = L2 − L3,
α3 = L1 − L3
How do we define the root spaces of the Cartan
Complex multiples of the E matrices
