module Math.Algebras.Quaternions where
import Math.Algebra.Field.Base
import Math.Algebras.VectorSpace
import Math.Algebras.TensorProduct
import Math.Algebras.Structures
data HBasis = One | I | J | K deriving (Eq,Ord)
type Quaternion k = Vect k HBasis
instance Show HBasis where
show One = "1"
show I = "i"
show J = "j"
show K = "k"
instance (Num k) => Algebra k HBasis where
unit 0 = zero
unit x = V [(One,x)]
mult = linear m
where m (One,b) = return b
m (b,One) = return b
m (I,I) = unit (1)
m (J,J) = unit (1)
m (K,K) = unit (1)
m (I,J) = return K
m (J,I) = 1 *> return K
m (J,K) = return I
m (K,J) = 1 *> return I
m (K,I) = return J
m (I,K) = 1 *> return J
i,j,k :: Num k => Quaternion k
i = return I
j = return J
k = return K
instance Num k => Coalgebra k HBasis where
counit (V ts) = sum [x | (One,x) <- ts]
comult = linear cm
where cm m = if m == One then return (m,m) else return (m,One) <+> return (One,m)