By Skoruppa N.-P.

Those are the notes of a path on Lie algebras which I gave on the collage of Bordeaux in spring 1997. The path used to be a so-called "Cours PostDEA", and as such needed to be held inside of 12 hours. much more difficult, no past wisdom approximately Lie algebras could be assumed. however, I had the target to arrive as height of the direction the nature formulation for Kac-Moody algebras, and, even as, to offer whole proofs so far as attainable.

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For a pair of roots α, β write (α, β) = |α| · |β| cos θ with a suitable 0 ≤ θ ≤ π. By axiom 4. we see that the numbers (α, β) (α, β) 4 cos2 θ = 2 ·2 |α|2 |β|2 are all integers. Since cos2 θ ≤ 1, we thus have only 5 possibilities for the value of 4 cos2 θ. Assume |α| ≥ |β| and α = ±β. Then the only possibilities for 4 cos2 θ are given by the following table: 4 cos2 θ 2 (α,β) 2 (β,α) |α|2 |β|2 0 0 0 1 1 1 1 −1 −1 2 1 2 2 −1 −2 3 1 3 3 −1 −3 |α|2 |β|2 θ ? 1 1 2 2 3 3 π 2 π 3 2π 3 π 4 3π 4 π 6 5π 6 Definition.

Indeed, take as basis xi the elements hα and elements zα ∈ Lα . Recall z α ∈ L−α , and [zα , z α ] = tα . Then, if α < 0, we have zα z α · v = 0, and if α > 0 we have zα z α · v = [zα , z α ] · v + z α zα · v = tα · v = (Λ, α)v. Thus, Ω·v = Λ(hα ) Λ(hα ) + α (Λ, α) v = (Λ + 2ρ, Λ) v. α 48 CHAPTER 4. 6). Since Ω commutes with L and since V = U (L)·vΛ , we find that Ω acts on a highest weight module with highest weight Λ as multiplication by (Λ, Λ+2ρ). But than it acts also as multiplication by (Λ, Λ + 2ρ) on any submodule, and on any quotient of such.

We show that it is invariant under W . e. the set of all β > 0 with nontrivial Lβ ). We remark first of all that, for any fundamental root α, the set R+ \ {α} is invariant under σα . Indeed, β ∈ R+ implies that σα (β) is a root, hence either negative or else positive. If σα (β) < 0, then σα (β) = β −kα for an integer k (= β(hα )) together with β > 0 imply that β is a multiple of α. But the only positive multiple of a fundamental root α which is a root, is α itself. Using that Lβ and Lσα (β) have same dimension, we now calculate σα D = 1 − e σα (−α) σα D 1 − e(−α) = 1 − e(α) D = −e(α) D.