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

Defined in Math.Algebras.LaurentPoly

Methods

munit :: LaurentMonomial

mmult :: LaurentMonomial -> LaurentMonomial -> LaurentMonomial

Mon [a]

Defined in Math.QuantumAlgebra.Tangle

Methods

munit :: [a]

mmult :: [a] -> [a] -> [a]

Ord v => Mon (Grevlex v)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

munit :: Grevlex v

mmult :: Grevlex v -> Grevlex v -> Grevlex v

Ord v => Mon (Glex v)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

munit :: Glex v

mmult :: Glex v -> Glex v -> Glex v

Ord v => Mon (Lex v)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

munit :: Lex v

mmult :: Lex v -> Lex v -> Lex v

Ord v => Mon (MonImpl v)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

munit :: MonImpl v

mmult :: MonImpl v -> MonImpl v -> MonImpl v

Ord a => Mon (SymmetricAlgebra a)

Defined in Math.Algebras.TensorAlgebra

Methods

munit :: SymmetricAlgebra a

mmult :: SymmetricAlgebra a -> SymmetricAlgebra a -> SymmetricAlgebra a

Mon (TensorAlgebra a)

Defined in Math.Algebras.TensorAlgebra

Methods

munit :: TensorAlgebra a

mmult :: TensorAlgebra a -> TensorAlgebra a -> TensorAlgebra a

Mon (NonComMonomial v)

Defined in Math.Algebras.NonCommutative

Methods

munit :: NonComMonomial v

mmult :: NonComMonomial v -> NonComMonomial v -> NonComMonomial v

(Mon a, Mon b) => Mon (Elim2 a b)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

munit :: Elim2 a b

mmult :: Elim2 a b -> Elim2 a b -> Elim2 a b

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

(Eq k, Num k) => Algebra k ()

Defined in Math.Algebras.Structures

Methods

unit :: k -> Vect k ()

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

(Eq k, Num k) => Algebra k HBasis

Defined in Math.Algebras.Quaternions

Methods

unit :: k -> Vect k HBasis

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

(Eq k, Num k) => Algebra k NSym

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k NSym

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

(Eq k, Num k) => Algebra k SymH

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k SymH

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

(Eq k, Num k) => Algebra k SymE

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k SymE

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

(Eq k, Num k) => Algebra k SymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k SymM

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

(Eq k, Num k) => Algebra k QSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k QSymF

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

(Eq k, Num k) => Algebra k QSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k QSymM

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

(Eq k, Num k) => Algebra k YSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k YSymM

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

(Eq k, Num k) => Algebra k SSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k SSymM

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

(Eq k, Num k) => Algebra k SSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k SSymF

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

(Eq k, Num k) => Algebra k OBasis

Defined in Math.Algebras.Octonions

Methods

unit :: k -> Vect k OBasis

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

(Eq k, Num k) => Algebra k M3

Defined in Math.Algebras.Matrix

Methods

unit :: k -> Vect k M3

mult :: Vect k (Tensor M3 M3) -> Vect k M3

(Eq k, Num k) => Algebra k Mat2

Defined in Math.Algebras.Matrix

Methods

unit :: k -> Vect k Mat2

mult :: Vect k (Tensor Mat2 Mat2) -> Vect k Mat2

(Eq k, Num k) => Algebra k LaurentMonomial

Defined in Math.Algebras.LaurentPoly

Methods

unit :: k -> Vect k LaurentMonomial

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

(Eq k, Num k) => Algebra k QNFBasis

Defined in Math.NumberTheory.QuadraticField

Methods

unit :: k -> Vect k QNFBasis

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

(Eq k, Num k, Ord b, Algebra k b) => Algebra k (Op b)

Defined in Math.Algebras.Structures

Methods

unit :: k -> Vect k (Op b)

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

(Eq k, Num k, Ord v, Show v) => Algebra k (Grevlex v)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

unit :: k -> Vect k (Grevlex v)

mult :: Vect k (Tensor (Grevlex v) (Grevlex v)) -> Vect k (Grevlex v)

(Eq k, Num k, Ord v, Show v) => Algebra k (Glex v)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

unit :: k -> Vect k (Glex v)

mult :: Vect k (Tensor (Glex v) (Glex v)) -> Vect k (Glex v)

(Eq k, Num k, Ord v, Show v) => Algebra k (Lex v)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

unit :: k -> Vect k (Lex v)

mult :: Vect k (Tensor (Lex v) (Lex v)) -> Vect k (Lex v)

(Eq k, Num k) => Algebra k (Permutation Int)

Defined in Math.Algebras.GroupAlgebra

Methods

unit :: k -> Vect k (Permutation Int)

mult :: Vect k (Tensor (Permutation Int) (Permutation Int)) -> Vect k (Permutation Int)

(Eq k, Num k, Ord a) => Algebra k (YSymF a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k (YSymF a)

mult :: Vect k (Tensor (YSymF a) (YSymF a)) -> Vect k (YSymF a)

(Eq k, Num k) => Algebra k (Dual SSymF)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k (Dual SSymF)

mult :: Vect k (Tensor (Dual SSymF) (Dual SSymF)) -> Vect k (Dual SSymF)

(Eq k, Num k, Ord a) => Algebra k (Shuffle a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

unit :: k -> Vect k (Shuffle a)

mult :: Vect k (Tensor (Shuffle a) (Shuffle a)) -> Vect k (Shuffle a)

(Eq k, Num k, Ord a) => Algebra k (Interval a)

The incidence algebra of a poset is the free k-vector space having as its basis the set of intervals in the poset, with multiplication defined by concatenation of intervals. The incidence algebra can also be thought of as the vector space of functions from intervals to k, with multiplication defined by the convolution (f*g)(x,y) = sum [ f(x,z) g(z,y) | x <= z <= y ].

Defined in Math.Combinatorics.IncidenceAlgebra

Methods

unit :: k -> Vect k (Interval a)

mult :: Vect k (Tensor (Interval a) (Interval a)) -> Vect k (Interval a)

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

Defined in Math.Algebras.TensorAlgebra

Methods

unit :: k -> Vect k (ExteriorAlgebra a)

mult :: Vect k (Tensor (ExteriorAlgebra a) (ExteriorAlgebra a)) -> Vect k (ExteriorAlgebra a)

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

Defined in Math.Algebras.TensorAlgebra

Methods

unit :: k -> Vect k (SymmetricAlgebra a)

mult :: Vect k (Tensor (SymmetricAlgebra a) (SymmetricAlgebra a)) -> Vect k (SymmetricAlgebra a)

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

Defined in Math.Algebras.TensorAlgebra

Methods

unit :: k -> Vect k (TensorAlgebra a)

mult :: Vect k (Tensor (TensorAlgebra a) (TensorAlgebra a)) -> Vect k (TensorAlgebra a)

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

Defined in Math.Algebras.NonCommutative

Methods

unit :: k -> Vect k (NonComMonomial v)

mult :: Vect k (Tensor (NonComMonomial v) (NonComMonomial v)) -> Vect k (NonComMonomial v)

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

Defined in Math.Algebras.Commutative

Methods

unit :: k -> Vect k (GlexMonomial v)

mult :: Vect k (Tensor (GlexMonomial v) (GlexMonomial v)) -> Vect k (GlexMonomial v)

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

Defined in Math.QuantumAlgebra.Tangle

Methods

unit :: k -> Vect k [a]

mult :: Vect k (Tensor [a] [a]) -> Vect k [a]

Algebra Q (SL2 ABCD)

Defined in Math.Algebras.AffinePlane

Methods

unit :: Q -> Vect Q (SL2 ABCD)

mult :: Vect Q (Tensor (SL2 ABCD) (SL2 ABCD)) -> Vect Q (SL2 ABCD)

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

The tensor product of k-algebras can itself be given the structure of a k-algebra

Defined in Math.Algebras.Structures

Methods

unit :: k -> Vect k (Tensor a b)

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

(Eq k, Num k, Ord a, Ord b, Algebra k a, Algebra k b) => Algebra k (DSum a b)

The direct sum of k-algebras can itself be given the structure of a k-algebra. This is the product object in the category of k-algebras.

Defined in Math.Algebras.Structures

Methods

unit :: k -> Vect k (DSum a b)

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

(Eq k, Num k, Ord a, Mon a, Ord b, Mon b) => Algebra k (Elim2 a b)

Defined in Math.CommutativeAlgebra.Polynomial

Methods

unit :: k -> Vect k (Elim2 a b)

mult :: Vect k (Tensor (Elim2 a b) (Elim2 a b)) -> Vect k (Elim2 a b)

Algebra (LaurentPoly Q) (SL2q String)

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

unit :: LaurentPoly Q -> Vect (LaurentPoly Q) (SL2q String)

mult :: Vect (LaurentPoly Q) (Tensor (SL2q String) (SL2q String)) -> Vect (LaurentPoly Q) (SL2q String)

Algebra (LaurentPoly Q) (M2q String)

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

unit :: LaurentPoly Q -> Vect (LaurentPoly Q) (M2q String)

mult :: Vect (LaurentPoly Q) (Tensor (M2q String) (M2q String)) -> Vect (LaurentPoly Q) (M2q String)

Algebra (LaurentPoly Q) (Aq02 String)

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

unit :: LaurentPoly Q -> Vect (LaurentPoly Q) (Aq02 String)

mult :: Vect (LaurentPoly Q) (Tensor (Aq02 String) (Aq02 String)) -> Vect (LaurentPoly Q) (Aq02 String)

Algebra (LaurentPoly Q) (Aq20 String)

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

unit :: LaurentPoly Q -> Vect (LaurentPoly Q) (Aq20 String)

mult :: Vect (LaurentPoly Q) (Tensor (Aq20 String) (Aq20 String)) -> Vect (LaurentPoly Q) (Aq20 String)

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

Sometimes it is more convenient to work with this version of unit.

class Coalgebra k b where

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

Methods

counit :: Vect k b -> k

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

Instances

(Eq k, Num k) => Coalgebra k EBasis

Defined in Math.Algebras.Structures

Methods

counit :: Vect k EBasis -> k

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

(Eq k, Num k) => Coalgebra k ()

Defined in Math.Algebras.Structures

Methods

counit :: Vect k () -> k

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

(Eq k, Num k) => Coalgebra k NSym

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k NSym -> k

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

(Eq k, Num k) => Coalgebra k SymH

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k SymH -> k

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

(Eq k, Num k) => Coalgebra k SymE

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k SymE -> k

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

(Eq k, Num k) => Coalgebra k SymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k SymM -> k

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

(Eq k, Num k) => Coalgebra k QSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k QSymF -> k

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

(Eq k, Num k) => Coalgebra k QSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k QSymM -> k

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

(Eq k, Num k) => Coalgebra k YSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k YSymM -> k

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

(Eq k, Num k) => Coalgebra k SSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k SSymM -> k

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

(Eq k, Num k) => Coalgebra k SSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k SSymF -> k

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

(Eq k, Num k) => Coalgebra k Mat2'

Defined in Math.Algebras.Matrix

Methods

counit :: Vect k Mat2' -> k

comult :: Vect k Mat2' -> Vect k (Tensor Mat2' Mat2')

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

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (MonoidCoalgebra m) -> k

comult :: Vect k (MonoidCoalgebra m) -> Vect k (Tensor (MonoidCoalgebra m) (MonoidCoalgebra m))

(Eq k, Num k) => Coalgebra k (SetCoalgebra b)

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (SetCoalgebra b) -> k

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

(Eq k, Num k, Ord b, Coalgebra k b) => Coalgebra k (Op b)

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (Op b) -> k

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

(Eq k, Num k) => Coalgebra k (Dual HBasis)

Defined in Math.Algebras.Quaternions

Methods

counit :: Vect k (Dual HBasis) -> k

comult :: Vect k (Dual HBasis) -> Vect k (Tensor (Dual HBasis) (Dual HBasis))

(Eq k, Num k) => Coalgebra k (Permutation Int)

Defined in Math.Algebras.GroupAlgebra

Methods

counit :: Vect k (Permutation Int) -> k

comult :: Vect k (Permutation Int) -> Vect k (Tensor (Permutation Int) (Permutation Int))

(Eq k, Num k, Ord a) => Coalgebra k (YSymF a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k (YSymF a) -> k

comult :: Vect k (YSymF a) -> Vect k (Tensor (YSymF a) (YSymF a))

(Eq k, Num k) => Coalgebra k (Dual SSymF)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k (Dual SSymF) -> k

comult :: Vect k (Dual SSymF) -> Vect k (Tensor (Dual SSymF) (Dual SSymF))

(Eq k, Num k, Ord a) => Coalgebra k (Shuffle a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

counit :: Vect k (Shuffle a) -> k

comult :: Vect k (Shuffle a) -> Vect k (Tensor (Shuffle a) (Shuffle a))

(Eq k, Num k, Ord a) => Coalgebra k (Interval a)

Defined in Math.Combinatorics.IncidenceAlgebra

Methods

counit :: Vect k (Interval a) -> k

comult :: Vect k (Interval a) -> Vect k (Tensor (Interval a) (Interval a))

(Eq k, Num k, Ord c) => Coalgebra k (TensorCoalgebra c)

Defined in Math.Algebras.TensorAlgebra

Methods

counit :: Vect k (TensorCoalgebra c) -> k

comult :: Vect k (TensorCoalgebra c) -> Vect k (Tensor (TensorCoalgebra c) (TensorCoalgebra c))

(Eq k, Num k) => Coalgebra k (GlexMonomial v)

Defined in Math.Algebras.Commutative

Methods

counit :: Vect k (GlexMonomial v) -> k

comult :: Vect k (GlexMonomial v) -> Vect k (Tensor (GlexMonomial v) (GlexMonomial v))

Coalgebra Q (SL2 ABCD)

Defined in Math.Algebras.AffinePlane

Methods

counit :: Vect Q (SL2 ABCD) -> Q

comult :: Vect Q (SL2 ABCD) -> Vect Q (Tensor (SL2 ABCD) (SL2 ABCD))

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

The tensor product of k-coalgebras can itself be given the structure of a k-coalgebra

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (Tensor a b) -> k

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

(Eq k, Num k, Ord a, Ord b, Coalgebra k a, Coalgebra k b) => Coalgebra k (DSum a b)

The direct sum of k-coalgebras can itself be given the structure of a k-coalgebra. This is the coproduct object in the category of k-coalgebras.

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (DSum a b) -> k

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

Coalgebra (LaurentPoly Q) (SL2q String)

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

counit :: Vect (LaurentPoly Q) (SL2q String) -> LaurentPoly Q

comult :: Vect (LaurentPoly Q) (SL2q String) -> Vect (LaurentPoly Q) (Tensor (SL2q String) (SL2q String))

Coalgebra (LaurentPoly Q) (M2q String)

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

counit :: Vect (LaurentPoly Q) (M2q String) -> LaurentPoly Q

comult :: Vect (LaurentPoly Q) (M2q String) -> Vect (LaurentPoly Q) (Tensor (M2q String) (M2q String))

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

Sometimes it is more convenient to work with this version of counit.

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

A bialgebra is an algebra which is also a coalgebra, subject to the compatibility conditions that counit and comult must be algebra morphisms (or equivalently, that unit and mult must be coalgebra morphisms)

Instances

(Eq k, Num k) => Bialgebra k NSym

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k SymH

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k SymE

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k SymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k QSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k QSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k YSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k SSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k SSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k (Permutation Int)

Defined in Math.Algebras.GroupAlgebra

(Eq k, Num k, Ord a) => Bialgebra k (YSymF a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k) => Bialgebra k (Dual SSymF)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

(Eq k, Num k, Ord a) => Bialgebra k (Shuffle a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Bialgebra Q (SL2 ABCD)

Defined in Math.Algebras.AffinePlane

Bialgebra (LaurentPoly Q) (SL2q String)

Defined in Math.QuantumAlgebra.QuantumPlane

Bialgebra (LaurentPoly Q) (M2q String)

Defined in Math.QuantumAlgebra.QuantumPlane

class Bialgebra k b => HopfAlgebra k b where

Methods

antipode :: Vect k b -> Vect k b

Instances

(Eq k, Num k) => HopfAlgebra k NSym

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k NSym -> Vect k NSym

(Eq k, Num k) => HopfAlgebra k SymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k SymM -> Vect k SymM

(Eq k, Num k) => HopfAlgebra k QSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k QSymF -> Vect k QSymF

(Eq k, Num k) => HopfAlgebra k QSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k QSymM -> Vect k QSymM

(Eq k, Num k) => HopfAlgebra k YSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k YSymM -> Vect k YSymM

(Eq k, Num k) => HopfAlgebra k SSymM

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k SSymM -> Vect k SSymM

(Eq k, Num k) => HopfAlgebra k SSymF

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k SSymF -> Vect k SSymF

(Eq k, Num k) => HopfAlgebra k (Permutation Int)

Defined in Math.Algebras.GroupAlgebra

Methods

antipode :: Vect k (Permutation Int) -> Vect k (Permutation Int)

(Eq k, Num k, Ord a) => HopfAlgebra k (YSymF a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k (YSymF a) -> Vect k (YSymF a)

(Eq k, Num k) => HopfAlgebra k (Dual SSymF)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k (Dual SSymF) -> Vect k (Dual SSymF)

(Eq k, Num k, Ord a) => HopfAlgebra k (Shuffle a)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

antipode :: Vect k (Shuffle a) -> Vect k (Shuffle a)

HopfAlgebra Q (SL2 ABCD)

Defined in Math.Algebras.AffinePlane

Methods

antipode :: Vect Q (SL2 ABCD) -> Vect Q (SL2 ABCD)

HopfAlgebra (LaurentPoly Q) (SL2q String)

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

antipode :: Vect (LaurentPoly Q) (SL2q String) -> Vect (LaurentPoly Q) (SL2q String)

newtype Op b

Constructors

Op b

Instances

(Eq k, Num k, Ord b, Coalgebra k b) => Coalgebra k (Op b)

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (Op b) -> k

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

(Eq k, Num k, Ord b, Algebra k b) => Algebra k (Op b)

Defined in Math.Algebras.Structures

Methods

unit :: k -> Vect k (Op b)

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

Eq b => Eq (Op b)

Defined in Math.Algebras.Structures

Methods

(==) :: Op b -> Op b -> Bool

(/=) :: Op b -> Op b -> Bool

Ord b => Ord (Op b)

Defined in Math.Algebras.Structures

Methods

compare :: Op b -> Op b -> Ordering

(<) :: Op b -> Op b -> Bool

(<=) :: Op b -> Op b -> Bool

(>) :: Op b -> Op b -> Bool

(>=) :: Op b -> Op b -> Bool

max :: Op b -> Op b -> Op b

min :: Op b -> Op b -> Op b

Show b => Show (Op b)

Defined in Math.Algebras.Structures

Methods

showsPrec :: Int -> Op b -> ShowS

show :: Op b -> String

showList :: [Op b] -> ShowS

newtype SetCoalgebra b

Constructors

SC b

Instances

(Eq k, Num k) => Coalgebra k (SetCoalgebra b)

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (SetCoalgebra b) -> k

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

Eq b => Eq (SetCoalgebra b)

Defined in Math.Algebras.Structures

Methods

(==) :: SetCoalgebra b -> SetCoalgebra b -> Bool

(/=) :: SetCoalgebra b -> SetCoalgebra b -> Bool

Ord b => Ord (SetCoalgebra b)

Defined in Math.Algebras.Structures

Methods

compare :: SetCoalgebra b -> SetCoalgebra b -> Ordering

(<) :: SetCoalgebra b -> SetCoalgebra b -> Bool

(<=) :: SetCoalgebra b -> SetCoalgebra b -> Bool

(>) :: SetCoalgebra b -> SetCoalgebra b -> Bool

(>=) :: SetCoalgebra b -> SetCoalgebra b -> Bool

max :: SetCoalgebra b -> SetCoalgebra b -> SetCoalgebra b

min :: SetCoalgebra b -> SetCoalgebra b -> SetCoalgebra b

Show b => Show (SetCoalgebra b)

Defined in Math.Algebras.Structures

Methods

showsPrec :: Int -> SetCoalgebra b -> ShowS

show :: SetCoalgebra b -> String

showList :: [SetCoalgebra b] -> ShowS

newtype MonoidCoalgebra m

Constructors

MC m

Instances

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

Defined in Math.Algebras.Structures

Methods

counit :: Vect k (MonoidCoalgebra m) -> k

comult :: Vect k (MonoidCoalgebra m) -> Vect k (Tensor (MonoidCoalgebra m) (MonoidCoalgebra m))

Eq m => Eq (MonoidCoalgebra m)

Defined in Math.Algebras.Structures

Methods

(==) :: MonoidCoalgebra m -> MonoidCoalgebra m -> Bool

(/=) :: MonoidCoalgebra m -> MonoidCoalgebra m -> Bool

Ord m => Ord (MonoidCoalgebra m)

Defined in Math.Algebras.Structures

Methods

compare :: MonoidCoalgebra m -> MonoidCoalgebra m -> Ordering

(<) :: MonoidCoalgebra m -> MonoidCoalgebra m -> Bool

(<=) :: MonoidCoalgebra m -> MonoidCoalgebra m -> Bool

(>) :: MonoidCoalgebra m -> MonoidCoalgebra m -> Bool

(>=) :: MonoidCoalgebra m -> MonoidCoalgebra m -> Bool

max :: MonoidCoalgebra m -> MonoidCoalgebra m -> MonoidCoalgebra m

min :: MonoidCoalgebra m -> MonoidCoalgebra m -> MonoidCoalgebra m

Show m => Show (MonoidCoalgebra m)

Defined in Math.Algebras.Structures

Methods

showsPrec :: Int -> MonoidCoalgebra m -> ShowS

show :: MonoidCoalgebra m -> String

showList :: [MonoidCoalgebra m] -> ShowS

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

Defined in Math.Algebras.Structures

Methods

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

(Eq k, Num k) => Module k Mat2 EBasis

Defined in Math.Algebras.Matrix

Methods

action :: Vect k (Tensor Mat2 EBasis) -> Vect k EBasis

(Eq k, 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)

Defined in Math.Algebras.Structures

Methods

action :: Vect k (Tensor a (Tensor u v)) -> Vect k (Tensor u v)

(Eq k, Num k) => Module k (Permutation Int) Int

Defined in Math.Algebras.GroupAlgebra

Methods

action :: Vect k (Tensor (Permutation Int) Int) -> Vect k Int

(Eq k, Num k) => Module k (Permutation Int) [Int]

Defined in Math.Algebras.GroupAlgebra

Methods

action :: Vect k (Tensor (Permutation Int) [Int]) -> Vect k [Int]

(Eq k, 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)

Defined in Math.Algebras.Structures

Methods

action :: Vect k (Tensor (Tensor a a) (Tensor u v)) -> Vect k (Tensor u v)

(*.) :: (Module k a m, Num k) => Vect k a -> Vect k m -> Vect k m

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

Defined in Math.Algebras.Structures

Methods

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

(Eq k, 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)

Defined in Math.Algebras.Structures

Methods

coaction :: Vect k (Tensor m n) -> Vect k (Tensor a (Tensor m n))

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

Defined in Math.QuantumAlgebra.QuantumPlane

Methods

coaction :: Vect (LaurentPoly Q) (Aq20 String) -> Vect (LaurentPoly Q) (Tensor (M2q String) (Aq20 String))

class HasPairing k u v where

A pairing is a non-degenerate bilinear form U x V -> k. We are typically interested in pairings having additional properties. For example:

• A bialgebra pairing is a pairing between bialgebras A and B such that the mult in A is adjoint to the comult in B, and vice versa, and the unit in A is adjoint to the counit in B, and vice versa.
• A Hopf pairing is a bialgebra pairing between Hopf algebras A and B such that the antipodes in A and B are adjoint.

Methods

pairing :: Vect k (Tensor u v) -> Vect k ()

Instances

(Eq k, Num k) => HasPairing k () ()

Defined in Math.Algebras.Structures

Methods

pairing :: Vect k (Tensor () ()) -> Vect k ()

(Eq k, Num k) => HasPairing k NSym QSymM

A duality pairing between NSym and QSymM (monomial basis), showing that NSym and QSym are dual.

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

pairing :: Vect k (Tensor NSym QSymM) -> Vect k ()

(Eq k, Num k) => HasPairing k SymH SymM

A duality pairing between the complete and monomial bases of Sym, showing that Sym is self-dual.

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

pairing :: Vect k (Tensor SymH SymM) -> Vect k ()

(Eq k, Num k) => HasPairing k SSymF SSymF

A pairing showing that SSym is self-adjoint

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

pairing :: Vect k (Tensor SSymF SSymF) -> Vect k ()

(Eq k, Num k) => HasPairing k SSymF (Dual SSymF)

Defined in Math.Combinatorics.CombinatorialHopfAlgebra

Methods

pairing :: Vect k (Tensor SSymF (Dual SSymF)) -> Vect k ()

(Eq k, Num k, HasPairing k u v, HasPairing k u' v') => HasPairing k (Tensor u u') (Tensor v v')

Defined in Math.Algebras.Structures

Methods

pairing :: Vect k (Tensor (Tensor u u') (Tensor v v')) -> Vect k ()

pairing' :: (Num k, HasPairing k u v) => Vect k u -> Vect k v -> k

The pairing function with a more Haskellish type signature

Orphan instances

(Eq k, Num k, Eq b, Ord b, Show b, Algebra k b) => Num (Vect k b)

Methods

(+) :: Vect k b -> Vect k b -> Vect k b

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

(*) :: Vect k b -> Vect k b -> Vect k b

negate :: Vect k b -> Vect k b

abs :: Vect k b -> Vect k b

signum :: Vect k b -> Vect k b

fromInteger :: Integer -> Vect k b