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Language	Haskell98

Numeric.Algebra.Hopf

Synopsis

class Bialgebra r h => HopfAlgebra r h where

Documentation

class Bialgebra r h => HopfAlgebra r h where #

A HopfAlgebra algebra on a semiring, where the module is free.

When antipode . antipode = id and antipode is an antihomomorphism then we are an InvolutiveBialgebra with inv = antipode as well

Minimal complete definition

antipode

Methods

antipode :: (h -> r) -> h -> r #

Instances

(Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k TrigBasis #
Instance details Defined in Numeric.Coalgebra.Trigonometric Methods antipode :: (TrigBasis -> k) -> TrigBasis -> k #
(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis' #
Instance details Defined in Numeric.Coalgebra.Quaternion Methods antipode :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r #
(Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis #
Instance details Defined in Numeric.Coalgebra.Hyperbolic Methods antipode :: (HyperBasis -> k) -> HyperBasis -> k #
(InvolutiveSemiring k, Rng k) => HopfAlgebra k DualBasis' #
Instance details Defined in Numeric.Coalgebra.Dual Methods antipode :: (DualBasis' -> k) -> DualBasis' -> k #
(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis #
Instance details Defined in Numeric.Algebra.Quaternion Methods antipode :: (QuaternionBasis -> r) -> QuaternionBasis -> r #
(Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis' #
Instance details Defined in Numeric.Algebra.Hyperbolic Methods antipode :: (HyperBasis' -> k) -> HyperBasis' -> k #
(InvolutiveSemiring k, Rng k) => HopfAlgebra k DualBasis #
Instance details Defined in Numeric.Algebra.Dual Methods antipode :: (DualBasis -> k) -> DualBasis -> k #
(InvolutiveSemiring k, Rng k) => HopfAlgebra k ComplexBasis #
Instance details Defined in Numeric.Algebra.Complex Methods antipode :: (ComplexBasis -> k) -> ComplexBasis -> k #
(HopfAlgebra r a, HopfAlgebra r b) => HopfAlgebra r (a, b) #
Instance details Defined in Numeric.Algebra.Hopf Methods antipode :: ((a, b) -> r) -> (a, b) -> r #
(HopfAlgebra r a, HopfAlgebra r b, HopfAlgebra r c) => HopfAlgebra r (a, b, c) #
Instance details Defined in Numeric.Algebra.Hopf Methods antipode :: ((a, b, c) -> r) -> (a, b, c) -> r #
(HopfAlgebra r a, HopfAlgebra r b, HopfAlgebra r c, HopfAlgebra r d) => HopfAlgebra r (a, b, c, d) #
Instance details Defined in Numeric.Algebra.Hopf Methods antipode :: ((a, b, c, d) -> r) -> (a, b, c, d) -> r #
(HopfAlgebra r a, HopfAlgebra r b, HopfAlgebra r c, HopfAlgebra r d, HopfAlgebra r e) => HopfAlgebra r (a, b, c, d, e) #
Instance details Defined in Numeric.Algebra.Hopf Methods antipode :: ((a, b, c, d, e) -> r) -> (a, b, c, d, e) -> r #