# Poisson Geometry
## Core Concepts
### Poisson Manifolds
1. **Poisson Bracket**
```math
\{f,g\}: C^\infty(M) \times C^\infty(M) \to C^\infty(M)
```
Properties:
- Antisymmetry: {f,g} = -{g,f}
- Leibniz: {f,gh} = {f,g}h + g{f,h}
- Jacobi: {f,{g,h}} + {g,{h,f}} + {h,{f,g}} = 0
2. **Poisson Bivector**
```math
π = \frac{1}{2}π^{ij}\frac{∂}{∂x^i} ∧ \frac{∂}{∂x^j}
```
where:
- π^{ij} are structure functions
- ∧ is wedge product
### Hamiltonian Systems
1. **Hamiltonian Vector Field**
```math
X_f = π^\sharp(df) = \{f,-\}
```
where:
- π^\sharp is sharp map
- df is differential
2. **Dynamics**
```math
\dot{x} = \{x,H\} = π^\sharp(dH)
```
where:
- H is Hamiltonian
- x are coordinates
## Advanced Concepts
### Poisson Reduction
1. **Momentum Map**
```math
J: M \to \mathfrak{g}^*, \quad ⟨J(x),ξ⟩ = J_ξ(x)
```
where:
- g* is Lie coalgebra
- J_ξ is moment function
2. **Reduced Bracket**
```math
\{f,g\}_{\text{red}}([x]) = \{F,G\}(x)
```
where:
- F,G are extensions
- [x] is equivalence class
### Symplectic Foliation
1. **Characteristic Distribution**
```math
C_x = \text{Im}(π^\sharp_x) \subset T_xM
```
where:
- π^\sharp is sharp map
- T_xM is tangent space
2. **Symplectic Leaves**
```math
(S,ω_S): \quad ω_S(X_f,X_g) = \{f,g\}
```
where:
- S is leaf
- ω_S is symplectic form
## Implementation
### Poisson Structure Computations
```python
class PoissonManifold:
def __init__(self,
dimension: int,
bivector: Callable):
"""Initialize Poisson manifold.
Args:
dimension: Manifold dimension
bivector: Poisson bivector field
"""
self.dim = dimension
self.pi = bivector
self._validate_jacobi_identity()
def poisson_bracket(self,
f: Callable,
g: Callable,
point: np.ndarray) -> float:
"""Compute Poisson bracket.
Args:
f,g: Functions on manifold
point: Evaluation point
Returns:
bracket: {f,g} at point
"""
# Get differentials
df = self._compute_differential(f, point)
dg = self._compute_differential(g, point)
# Contract with bivector
return self._contract_with_bivector(df, dg, point)
def hamiltonian_vector_field(self,
f: Callable,
point: np.ndarray) -> np.ndarray:
"""Compute Hamiltonian vector field.
Args:
f: Function on manifold
point: Base point
Returns:
X_f: Vector field at point
"""
# Get differential
df = self._compute_differential(f, point)
# Apply sharp map
return self._sharp_map(df, point)
```
### Reduction Theory
```python
class PoissonReduction:
def __init__(self,
poisson_manifold: PoissonManifold,
group_action: GroupAction):
"""Initialize Poisson reduction.
Args:
poisson_manifold: Manifold
group_action: Lie group action
"""
self.manifold = poisson_manifold
self.action = group_action
def momentum_map(self,
point: np.ndarray) -> np.ndarray:
"""Compute momentum map.
Args:
point: Point on manifold
Returns:
J: Momentum map value
"""
# Get infinitesimal generators
generators = self.action.get_generators(point)
# Compute moment functions
return self._compute_momentum(generators, point)
def reduce(self,
level: np.ndarray) -> PoissonManifold:
"""Perform Poisson reduction.
Args:
level: Momentum level
Returns:
reduced: Reduced Poisson manifold
"""
# Get level set
level_set = self._compute_level_set(level)
# Compute quotient
return self._compute_quotient(level_set)
```
## Advanced Applications
### Deformation Quantization
```python
class DeformationQuantization:
def __init__(self,
poisson_manifold: PoissonManifold,
formal_parameter: Symbol):
"""Initialize deformation quantization.
Args:
poisson_manifold: Classical phase space
formal_parameter: Planck constant
"""
self.manifold = poisson_manifold
self.hbar = formal_parameter
def star_product(self,
f: Callable,
g: Callable,
order: int) -> Expression:
"""Compute star product.
Args:
f,g: Functions to multiply
order: Expansion order
Returns:
product: f * g up to order
"""
# Compute bidifferential operators
operators = self._compute_bidifferential_operators(order)
# Sum expansion
return self._sum_star_product(f, g, operators)
def verify_associativity(self,
f: Callable,
g: Callable,
h: Callable,
order: int) -> bool:
"""Verify associativity equations.
Args:
f,g,h: Test functions
order: Verification order
Returns:
associative: Whether equations hold
"""
return self._check_associativity(f, g, h, order)
```
### Quantum Groups
```python
class QuantumGroup:
def __init__(self,
lie_bialgebra: LieBialgebra,
deformation_parameter: Symbol):
"""Initialize quantum group.
Args:
lie_bialgebra: Classical structure
deformation_parameter: q-parameter
"""
self.bialgebra = lie_bialgebra
self.q = deformation_parameter
def quantize_structure(self) -> HopfAlgebra:
"""Quantize Lie bialgebra.
Returns:
quantum: Quantum group
"""
# Compute universal R-matrix
R = self._compute_R_matrix()
# Construct coproduct
Delta = self._construct_coproduct(R)
# Build Hopf algebra
return self._construct_hopf_algebra(Delta, R)
def classical_limit(self,
element: AlgebraElement) -> LieElement:
"""Compute classical limit.
Args:
element: Quantum group element
Returns:
classical: Classical limit
"""
return self._take_classical_limit(element)
```
## Advanced Topics
### Lie Bialgebras
1. **Cobracket**
```math
δ: \mathfrak{g} \to \mathfrak{g} ∧ \mathfrak{g}
```
where:
- g is Lie algebra
- δ is 1-cocycle
2. **Double**
```math
\mathfrak{d} = \mathfrak{g} ⊕ \mathfrak{g}^*
```
where:
- g* is dual algebra
- d is double
### Integrable Systems
1. **Casimir Functions**
```math
\{C,f\} = 0 \text{ for all } f
```
where:
- C is Casimir
- {,} is Poisson bracket
2. **Bi-Hamiltonian Structure**
```math
π_1, π_2: \quad [π_1,π_2] = 0
```
where:
- π_i are Poisson structures
- [,] is Schouten bracket
## Future Directions
### Emerging Areas
1. **Higher Structures**
- Higher Poisson Geometry
- Derived Poisson Geometry
- Graded Poisson Manifolds
2. **Applications**
- Quantum Field Theory
- String Theory
- Noncommutative Geometry
### Open Problems
1. **Theoretical Challenges**
- Formality Conjecture
- Integration Problem
- Quantization Program
2. **Computational Challenges**
- Star Product Construction
- R-matrix Computation
- Deformation Theory
## Related Topics
1. [[symplectic_geometry|Symplectic Geometry]]
2. [[quantum_groups|Quantum Groups]]
3. [[deformation_quantization|Deformation Quantization]]
4. [[integrable_systems|Integrable Systems]]