Math.Algebras.Structures

A module defining various algebraic structures that can be defined on vector spaces - specifically algebra, coalgebra, bialgebra, Hopf algebra, module, comodule

Documentation

class Mon m where

Monoid

Methods

munit :: m

mmult :: m -> m -> m

Instances

Mon LaurentMonomial

Mon [a]

Mon (Permutation Int)

Mon (NonComMonomial v)

Ord a => Mon (SymmetricAlgebra a)

Mon (TensorAlgebra a)

class Algebra k b where

Caution: If we declare an instance Algebra k b, then we are saying that the vector space Vect k b is a k-algebra. In other words, we are saying that b is the basis for a k-algebra. So a more accurate name for this class would have been AlgebraBasis.

Methods

unit :: k -> Vect k b

mult :: Vect k (Tensor b b) -> Vect k b

Instances

Num k => Algebra k ()

Num k => Algebra k LaurentMonomial

Num k => Algebra k M3

Num k => Algebra k Mat2

Num k => Algebra k HBasis

(Num k, Ord v) => Algebra k (GlexMonomial v)

Num k => Algebra k (Permutation Int)

(Num k, Ord v) => Algebra k (NonComMonomial v)

(Num k, Ord a) => Algebra k (ExteriorAlgebra a)

(Num k, Ord a) => Algebra k (SymmetricAlgebra a)

(Num k, Ord a) => Algebra k (TensorAlgebra a)

(Num k, Ord a) => Algebra k [a]

Algebra Q (SL2 ABCD)

(Num k, Ord a, Ord b, Algebra k a, Algebra k b) => Algebra k (Tensor a b)

Algebra (LaurentPoly Q) (SL2q String)

Algebra (LaurentPoly Q) (M2q String)

Algebra (LaurentPoly Q) (Aq02 String)

Algebra (LaurentPoly Q) (Aq20 String)

class Coalgebra k b where

An instance declaration for Coalgebra k b is saying that the vector space Vect k b is a k-algebra.

Methods

counit :: Vect k b -> k

comult :: Vect k b -> Vect k (Tensor b b)

Instances

Num k => Coalgebra k ()

Num k => Coalgebra k Mat2'

Num k => Coalgebra k HBasis

(Num k, Ord m, Mon m) => Coalgebra k (MonoidCoalgebra m)

Num k => Coalgebra k (SetCoalgebra b)

Num k => Coalgebra k (Permutation Int)

Coalgebra Q (SL2 ABCD)

(Num k, Ord a, Ord b, Coalgebra k a, Coalgebra k b) => Coalgebra k (Tensor a b)

Coalgebra (LaurentPoly Q) (SL2q String)

Coalgebra (LaurentPoly Q) (M2q String)

class (Algebra k b, Coalgebra k b) => Bialgebra k b

A bialgebra is an algebra which is also a coalgebra, subject to some compatibility conditions

Instances

Num k => Bialgebra k (Permutation Int)

Bialgebra Q (SL2 ABCD)

Bialgebra (LaurentPoly Q) (SL2q String)

Bialgebra (LaurentPoly Q) (M2q String)

class Bialgebra k b => HopfAlgebra k b where

Methods

antipode :: Vect k b -> Vect k b

Instances

Num k => HopfAlgebra k (Permutation Int)

HopfAlgebra Q (SL2 ABCD)

HopfAlgebra (LaurentPoly Q) (SL2q String)

type Trivial k = Vect k ()

Trivial k is the field k considered as a k-vector space. In maths, we would not normally make a distinction here, but in the code, we need this if we want to be able to put k as one side of a tensor product.

unit' :: (Num k, Algebra k b) => Trivial k -> Vect k b

counit' :: (Num k, Coalgebra k b) => Vect k b -> Trivial k

newtype SetCoalgebra b

Constructors

SC b

Instances

Num k => Coalgebra k (SetCoalgebra b)

Eq b => Eq (SetCoalgebra b)

Ord b => Ord (SetCoalgebra b)

Show b => Show (SetCoalgebra b)

newtype MonoidCoalgebra m

Constructors

MC m

Instances

(Num k, Ord m, Mon m) => Coalgebra k (MonoidCoalgebra m)

Eq m => Eq (MonoidCoalgebra m)

Ord m => Ord (MonoidCoalgebra m)

Show m => Show (MonoidCoalgebra m)

class Algebra k a => Module k a m where

Methods

action :: Vect k (Tensor a m) -> Vect k m

Instances

Algebra k a => Module k a a

Num k => Module k Mat2 EBasis

(Num k, Ord a, Ord u, Ord v, Bialgebra k a, Module k a u, Module k a v) => Module k a (Tensor u v)

Num k => Module k (Permutation Int) Int

(Num k, Ord a, Ord u, Ord v, Algebra k a, Module k a u, Module k a v) => Module k (Tensor a a) (Tensor u v)

class Coalgebra k c => Comodule k c n where

Methods

coaction :: Vect k n -> Vect k (Tensor c n)

Instances

Coalgebra k c => Comodule k c c

(Num k, Ord a, Ord m, Ord n, Bialgebra k a, Comodule k a m, Comodule k a n) => Comodule k a (Tensor m n)

Comodule (LaurentPoly Q) (M2q String) (Aq20 String)