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4.2 Roots

The adjoint action of the Cartan subalgebra on 𝔫+ and 𝔫− is diagonal. Explicitly,
[h,ei] = αi(h )ei (no summation on i) (4.14 )
for any element h ∈ 𝔥, where α i is the linear form on 𝔥 (i.e., the element of the dual 𝔥 ∗) defined by αi(hj) = Aji. The αi’s are called the simple roots. Similarly,
[h, [e ,[e ,⋅⋅⋅ ,[e ,e ]⋅⋅⋅]] = (α + α + ⋅⋅⋅α )(h)[e ,[e ,⋅⋅⋅ ,[e ,e ]⋅⋅⋅] (4.15 ) i1 i2 ik−1 ik i1 i2 ik i1 i2 ik−1 ik
and, if [ei1,[ei2,⋅⋅⋅ ,[eik−1,eik]⋅⋅⋅] is non-zero, one says that αi1 + αi2 + ⋅⋅⋅αik is a positive root. On the negative side, 𝔫−, one has
[h,[f ,[f ,⋅⋅⋅ ,[f ,f ]⋅⋅⋅]] = − (α + α + ⋅⋅⋅α )(h)[f ,[f ,⋅⋅⋅ ,[f ,f ]⋅⋅⋅] (4.16 ) i1 i2 ik−1 ik i1 i2 ik i1 i2 ik−1 ik
and − (αi1 + αi2 + ⋅⋅⋅ αik)(h) is called a negative root when [fi1,[fi2,⋅⋅⋅ ,[fik−1,fik] is non-zero. This occurs if and only if [ei,[ei ,⋅⋅⋅ ,[ei ,ei ]⋅⋅⋅] 1 2 k−1 k is non-zero: − α is a negative root if and only if α is a positive root.

We see from the construction that the roots (linear forms α such that [h, x] = α (h)x has nonzero solutions x) are either positive (linear combinations of the simple roots αi with integer non-negative coefficients) or negative (linear combinations of the simple roots with integer non-positive coefficients). The set of positive roots is denoted by Δ +; that of negative roots by Δ −. The set of all roots is Δ, so we have Δ = Δ+ ∪ Δ −. The simple roots are positive and form a basis of ∗ 𝔥. One sometimes denotes the hi by ∨ αi (and thus, ∨ [α i ,ej] = Aijej etc). Similarly, one also uses the notation ⟨⋅,⋅⟩ for the standard pairing between 𝔥 and its dual 𝔥∗, i.e., ⟨α, h⟩ = α(h). In this notation the entries of the Cartan matrix can be written as

∨ ∨ Aij = αj(αi ) = ⟨αj, αi ⟩ . (4.17 )

Finally, the root lattice Q is the set of linear combinations with integer coefficients of the simple roots,

∑ Q = ℤ αi. (4.18 ) i

All roots belong to the root lattice, of course, but the converse is not true: There are elements of Q that are not roots.

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