[x,y] bracket, Jacobi. sl₂ = {e,h,f}. Adjoint representation. Finite-dim irreducible sl₂-modules: highest weight λ, dim λ+1.
graph TD
D1["Def: Lie algebra\n[x,y] bilinear, Jacobi"]
D2["Def: sl₂\ne,h,f brackets"]
D3["Def: Adjoint\nad x y = [x,y]"]
D4["Def: Weight space\nV_μ = ker h−μ"]
T1["Thm: sl₂ irreps\nV_n dim n+1"]
T2["Thm: Highest weight\nv, ev=0, hv=λv"]
T3["Thm: v,fv,f²v... basis"]
T4["Thm: Casimir\nC central"]
T5["Thm: Weyl unitary trick\nsl₂ℂ reps"]
L1["Lemma: [e,f]=h"]
D1 --> D2
D2 --> L1
D2 --> D3
D4 --> T2
T2 --> T1
T1 --> T3
D2 --> T4
T1 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 15
- Edges: 12
- Definitions: 4
- Theorems: 5
- Lemmas: 1
Frontier: Kac–Moody, quantum groups, geometric Langlands. math.RT