Math.Manifold.Homogeneous

Documentation

class (Semimanifold g, Monoid g) => LieGroup g where

Manifolds with a continuous group structure, whose Needle space is isomorphic to the associated Lie algebra.

Laws:

  expMap zeroV ≡ mempty
  lieBracket w v ≡ negateV (lieBracket v w)
  ...
  

Methods

expMap :: LieAlgebra g -> g

lieBracket :: Bilinear (LieAlgebra g) (LieAlgebra g) (LieAlgebra g)

data LieAlgebra g

Instances

Semimanifold g => AbstractAdditiveGroup (LieAlgebra g)

Defined in Math.Manifold.Homogeneous

Associated Types

type VectorSpaceImplementation (LieAlgebra g)

Semimanifold g => AdditiveGroup (LieAlgebra g)

Defined in Math.Manifold.Homogeneous

Methods

zeroV :: LieAlgebra g

(^+^) :: LieAlgebra g -> LieAlgebra g -> LieAlgebra g

negateV :: LieAlgebra g -> LieAlgebra g

(^-^) :: LieAlgebra g -> LieAlgebra g -> LieAlgebra g

type VectorSpaceImplementation (LieAlgebra g)

Defined in Math.Manifold.Homogeneous

type VectorSpaceImplementation (LieAlgebra g) = Needle g

class (Semimanifold m, LieGroup g) => ActsOn g m where

Manifolds that are homogeneous with respect to action by a Lie group. Laws:

  action mempty ≡ id                  (Identity)
  action (a<>b) ≡ action a . action b (Compatibility)
  

Methods

action :: g -> m -> m

type SO n = SO_ n Double