Poisson Structures

graph TD D1["D1 Poisson bracket π\nSkew-symmetric, Leibniz, Jacobi identity"] D2["D2 Poisson manifold (M, π)\nBivector field with [π,π] = 0"] D3["D3 Symplectic leaf\nMaximal integral submanifold of π#(T*M)"] D4["D4 Casimir function C\n{C, f} = 0 for all f"] T1["T1 ω symplectic ⇒ {f,g} = ω(X_f,X_g)\nDefines Poisson"] T2["T2 Symplectic foliation\n(M,π) union of symplectic leaves"] T3["T3 Darboux-Weinstein\nLocal splitting: symplectic × cosymplectic"] T4["T4 Kostant-Souriau\nPrequantization of integral ω"] T5["T5 Kontsevich formality\nDeformation quantization exists"] D1 --> D2 D2 --> D3 D2 --> D4 D1 --> T1 D3 --> T2 D2 --> T3 D4 --> T2 T1 --> D2 T2 --> T3 T3 --> T5 classDef definition fill:#3498db,color:#fff,stroke:#2980b9 classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 class D1,D2,D3,D4 definition class T1,T2,T3,T4,T5 theorem