Lie.LieAlgebra

This module translates the mathematical definition of a Lie Algebra into a Class LieAlgebra, which also wraps around the other modules.

Documentation

class LieAlgebra a s | a -> s where

Vectorspace with Lie Bracket

Minimal complete definition

(|+|), (|*|), (|.|), basis, trace

Methods

(|+|) :: a -> a -> a

Addition should satisfy the following attributes.

• Associativity:

(x |+| y) |+| z == x |+| (y |+| z)

• Commutativity:

x |+| y == y |+| x

• Neutral element 0 exists:

x |+| 0 == x

• Inverse element (-x) exists:

x |+| (-x) == 0

(|*|) :: s -> a -> a

Scalar Multiplication should satisfy the following attributes.

• Distributivity:

a |*| (x |+| y) == (a |*| x) |+| (a |*| y)

• Neutral element 1 exists:

1 |*| x == x

(|.|) :: a -> a -> a

Lie Bracket should satisfy the following attributes.

• Bilinearity:

a |*| (x |.| y) == (a |*| x) |.| y

(x |+| y) |.| z == (x |.| z) |+| (y |.| z)

• Antisymmetry:

x |.| y == - (y |.| x)

• Jacobi-Identity:

x |.| (y |.| z) + y |.| (z |.| x) + z |.| (x |.| y) = 0

basis :: [a]

Ordered Basis Vectors of the Lie Algebra:

• linearly independent
• span the whole Vectorspace

linearCombination :: [s] -> a

Linear Combination of basis vectors, where the order refers to the Basis and the values are the coefficients

adjunction :: a -> a -> a

Natural way one would define a dual to the elements in the Lie Algebra

trace :: (a -> a) -> s

Calculates the Trace of an object in the Dual Vectorspace of the Lie Algebra

(<|>) :: (a -> a) -> (a -> a) -> s

Killing Form, which is a scalar product on the Dual Vectorspace of the Lie Algebra

Instances

LieAlgebra Vector Double

Defined in LieExample.SO3

Methods

(|+|) :: Vector -> Vector -> Vector

(|*|) :: Double -> Vector -> Vector

(|.|) :: Vector -> Vector -> Vector

basis :: [Vector]

linearCombination :: [Double] -> Vector

adjunction :: Vector -> Vector -> Vector

trace :: (Vector -> Vector) -> Double

(<|>) :: (Vector -> Vector) -> (Vector -> Vector) -> Double

LieAlgebra Matrix CD

Defined in LieExample.SU3

Methods

(|+|) :: Matrix -> Matrix -> Matrix

(|*|) :: CD -> Matrix -> Matrix

(|.|) :: Matrix -> Matrix -> Matrix

basis :: [Matrix]

linearCombination :: [CD] -> Matrix

adjunction :: Matrix -> Matrix -> Matrix

trace :: (Matrix -> Matrix) -> CD

(<|>) :: (Matrix -> Matrix) -> (Matrix -> Matrix) -> CD