La categorie O des représentations d`une algèbre de Lie
=ℝ ℂ
−𝔤−
𝔤−( ) [ ]
[ [ ]] + [ [ ]] + [ [ ]] = 0∀ ∈
−
[ ] := − ∀ ∈
()
𝔤−
−:𝔤7→ ()
∀ ∈ ([ ]) = ()()−()()
𝔤()
𝔤−
𝔰𝔩𝑛() := {∈𝑛() = 0}
𝔰𝔩𝑛→
𝑛𝑛: [ ] =
− 𝑛𝔰𝔩𝑛−
𝔰𝔩𝑛
𝔤
(𝔤) =
𝑛≥0
𝔤⊗𝑛
𝔤⊗0=
𝔤
(𝔤) = (𝔤) ⊗−⊗−[ ]
−
𝑘−𝐿𝑖𝑒(𝔤 ) = 𝑘−𝑎𝑙𝑔((𝔤) )
𝔤(𝔤)−
−𝔤
(𝔤)−
𝔤−(𝔤)−
(𝔤)𝔤
(1 𝑟)𝔤
𝜈1
1 𝜈𝑟
𝑟(1 𝑟)∈ℕ𝑟
(𝔤)
𝔫+𝔤𝔫−𝔤
𝔤=𝔫−⊕𝔥⊕𝔫+
(𝔤) = (𝔫−)⊗(𝔤)⊗(𝔫+)
(𝔤)
(𝑛)𝑛𝑛
(𝔤)→(𝔤)
𝐹(𝔤) = ⊕𝑛𝑛+1𝑛
(𝔤)
(𝔤)𝐹(𝔤)
(𝔤)
(𝔤)□
0
ℂ
𝔤
𝕂−∈𝑘()
∈𝑘()∈ℕ𝑛= 0
∈𝔤(𝑠 𝑛)∈𝔤×𝔤=𝑠+𝑛
=𝑠+𝑛𝑠𝑛
𝑠 𝑛
𝔱 𝔤
− 𝒞
𝒞
𝒟
𝒟 𝒞
𝒞
∈ 𝒞
∈ := (−)
𝒞1∈
𝒞 𝒞 1
𝒞
𝒟
∈ 𝒟 =1⊕ ⋅ ⋅ ⋅ ⊕ 𝑟
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