A module defining various algebraic structures that can be defined on vector spaces - specifically algebra, coalgebra, bialgebra, Hopf algebra, module, comodule
- class Mon m where
- class Algebra k b where
- class Coalgebra k b where
- class (Algebra k b, Coalgebra k b) => Bialgebra k b
- class Bialgebra k b => HopfAlgebra k b where
- 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 = SC b
- newtype MonoidCoalgebra m = MC m
- class Algebra k a => Module k a m where
- class Coalgebra k c => Comodule k c n where
Documentation
Monoid
Mon LaurentMonomial | |
Mon [a] | |
Mon (Permutation Int) | |
Mon (NonComMonomial v) | |
Ord a => Mon (SymmetricAlgebra a) | |
Mon (TensorAlgebra a) |
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.
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] | |
(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 ]. |
Algebra Q (SL2 ABCD) | |
(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 |
(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. |
Algebra (LaurentPoly Q) (SL2q String) | |
Algebra (LaurentPoly Q) (M2q String) | |
Algebra (LaurentPoly Q) (Aq02 String) | |
Algebra (LaurentPoly Q) (Aq20 String) |
class Coalgebra k b whereSource
An instance declaration for Coalgebra k b is saying that the vector space Vect k b is a k-algebra.
Num k => Coalgebra k EBasis | |
Num k => Coalgebra k () | |
Num k => Coalgebra k Mat2' | |
(Num k, Ord m, Mon m) => Coalgebra k (MonoidCoalgebra m) | |
Num k => Coalgebra k (SetCoalgebra b) | |
Num k => Coalgebra k (GlexMonomial v) | |
Num k => Coalgebra k (Permutation Int) | |
Num k => Coalgebra k (Dual HBasis) | |
(Num k, Ord c) => Coalgebra k (TensorCoalgebra c) | |
(Num k, Ord a) => Coalgebra k (Interval a) | |
Coalgebra Q (SL2 ABCD) | |
(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 |
(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. |
Coalgebra (LaurentPoly Q) (SL2q String) | |
Coalgebra (LaurentPoly Q) (M2q String) |
class (Algebra k b, Coalgebra k b) => Bialgebra k b Source
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)
class Bialgebra k b => HopfAlgebra k b whereSource
Num k => HopfAlgebra k (Permutation Int) | |
HopfAlgebra Q (SL2 ABCD) | |
HopfAlgebra (LaurentPoly Q) (SL2q String) |
newtype SetCoalgebra b Source
SC b |
Num k => Coalgebra k (SetCoalgebra b) | |
Eq b => Eq (SetCoalgebra b) | |
Ord b => Ord (SetCoalgebra b) | |
Show b => Show (SetCoalgebra b) |
newtype MonoidCoalgebra m Source
MC m |
(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 whereSource
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) |