HaskellForMaths-0.4.5: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Algebras.VectorSpace

Description

A module defining the type and operations of free k-vector spaces over a basis b (for a field k)

Synopsis

# Documentation

newtype Vect k b Source

Given a field type k and a basis type b, Vect k b is the type of the free k-vector space over b. Elements (values) of Vect k b consist of k-linear combinations of elements (values) of b.

In order for Vect k b to be a vector space, it is necessary that k is a field (that is, an instance of Fractional). In practice, we often relax this condition, and require that k is a ring (that is, an instance of Num). In that case, Vect k b should more correctly be called (the type of) the free k-module over b.

Most of the code requires that b is an instance of Ord. This is primarily to enable us to simplify to a normal form.

Constructors

 V [(b, k)]

Instances

 Fractional QNF (Eq k, Fractional k) => Fractional (LaurentPoly k) Num k => Monad (Vect k) Given a field k, the type constructor (Vect k) is a monad, the "free k-vector space monad". In order to understand this, it is probably easiest to think of a free k-vector space as a kind of container, a bit like a list, except that order doesn't matter, and you're allowed arbitrary (even negative or fractional) quantities of the basis elements in the container. According to this way of thinking, return is the function that puts a basis element into the vector space (container). Given a function f from the basis of one vector space to another vector space (a -> Vect k b), bind (>>=) lifts it to a function (>>= f) from the first vector space to the second (Vect k a -> Vect k b). Note that in general (>>= f) applied to a vector will not return a result in normal form, so it is usually preferable to use (linear f) instead. Functor (Vect k) Given a field k, (Vect k) is a functor, the "free k-vector space" functor. In the mathematical sense, this can be regarded as a functor from the category Set (of sets) to the category k-Vect (of k-vector spaces). In Haskell, instead of Set we have Hask, the category of Haskell types. However, for our purposes it is helpful to identify Hask with Set, but identifying a Haskell type with its set of inhabitants. The type constructor (Vect k) gives the action of the functor on objects in the category, taking a set (type) to a free k-vector space. fmap gives the action of the functor on arrows in the category, taking a function between sets (types) to a linear map between vector spaces. Note that if f is not order-preserving, then (fmap f) is not guaranteed to return results in normal form, so it may be preferable to use (nf . fmap f). HasInverses (GroupAlgebra Q) Note that the inverse of a group algebra element can only be efficiently calculated if the group generated by the non-zero terms is very small (eg <100 elements). HopfAlgebra (LaurentPoly Q) (SL2q String) Bialgebra (LaurentPoly Q) (SL2q String) Bialgebra (LaurentPoly Q) (M2q String) Coalgebra (LaurentPoly Q) (SL2q String) Coalgebra (LaurentPoly Q) (M2q String) Algebra (LaurentPoly Q) (SL2q String) Algebra (LaurentPoly Q) (M2q String) Algebra (LaurentPoly Q) (Aq02 String) Algebra (LaurentPoly Q) (Aq20 String) Comodule (LaurentPoly Q) (M2q String) (Aq20 String) (Eq k, Eq b) => Eq (Vect k b) (Eq k, Fractional k, Monomial m, Ord m, Algebra k m) => Fractional (Vect k m) As a convenience, a partial instance of Fractional is defined for polynomials. The instance is well-defined only for scalars, and gives an error if used on other values. The purpose of this is to allow entry of fractional scalars, in expressions such as `x/2`. On the other hand, an expression such as `2/x` will return an error. (Eq k, Fractional k, Ord a, Show a, HasConjugation k a) => Fractional (Vect k a) If an algebra has a conjugation operation, then it has multiplicative inverses, via 1x = conj x sqnorm x (Eq k, Num k, Eq b, Ord b, Show b, Algebra k b) => Num (Vect k b) (Ord k, Ord b) => Ord (Vect k b) (Show k, Eq k, Num k, Show b) => Show (Vect k b) (Eq k, Fractional k, Ord a, Show a) => HasInverses (Vect k (Interval a))

coeff :: (Eq a1, Num a) => a1 -> Vect a a1 -> aSource

Return the coefficient of the specified basis element in a vector

removeTerm :: (Eq k, Num k, Ord a) => a -> Vect k a -> Vect k aSource

Remove the term for a specified basis element from a vector

zerov :: Vect k bSource

The zero vector

add :: (Ord b, Eq k, Num k) => Vect k b -> Vect k b -> Vect k bSource

(<+>) :: (Ord b, Eq k, Num k) => Vect k b -> Vect k b -> Vect k bSource

sumv :: (Ord b, Eq k, Num k) => [Vect k b] -> Vect k bSource

Sum of a list of vectors

negatev :: (Eq k, Num k) => Vect k b -> Vect k bSource

Negation of a vector

(<->) :: (Ord b, Eq k, Num k) => Vect k b -> Vect k b -> Vect k bSource

Subtraction of vectors

smultL :: (Eq k, Num k) => k -> Vect k b -> Vect k bSource

Scalar multiplication (on the left)

(*>) :: (Eq k, Num k) => k -> Vect k b -> Vect k bSource

Same as smultL. Mnemonic is "multiply through (from the left)"

smultR :: (Eq k, Num k) => Vect k b -> k -> Vect k bSource

Scalar multiplication on the right

(<*) :: (Eq k, Num k) => Vect k b -> k -> Vect k bSource

Same as smultR. Mnemonic is "multiply through (from the right)"

nf :: (Ord b, Eq k, Num k) => Vect k b -> Vect k bSource

Convert an element of Vect k b into normal form. Normal form consists in having the basis elements in ascending order, with no duplicates, and all coefficients non-zero

linear :: (Ord b, Eq k, Num k) => (a -> Vect k b) -> Vect k a -> Vect k bSource

A linear map between vector spaces A and B can be defined by giving its action on the basis elements of A. The action on all elements of A then follows by linearity.

If we have A = Vect k a, B = Vect k b, and f :: a -> Vect k b is a function from the basis elements of A into B, then `linear f` is the linear map that this defines by linearity.

type Trivial k = Vect k ()Source

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.

wrap :: (Eq k, Num k) => k -> Vect k ()Source

Wrap an element of the field k to an element of the trivial k-vector space

unwrap :: Num k => Vect k () -> kSource

Unwrap an element of the trivial k-vector space to an element of the field k

newtype Dual b Source

Given a finite vector space basis b, Dual b can be used to represent a basis for the dual vector space. The intention is that for a given individual basis element b_i, (Dual b_i) represents the indicator function for b_i, which takes b_i to 1 and all other basis elements to 0.

(Note that if the basis b is infinite, then Dual b may only represent a sub-basis of the dual vector space.)

Constructors

 Dual b

Instances

 (Eq k, Num k) => HasPairing k SSymF (Dual SSymF) (Eq k, Num k) => HopfAlgebra k (Dual SSymF) (Eq k, Num k) => Bialgebra k (Dual SSymF) (Eq k, Num k) => Coalgebra k (Dual HBasis) (Eq k, Num k) => Coalgebra k (Dual SSymF) (Eq k, Num k) => Algebra k (Dual SSymF) Eq b => Eq (Dual b) Ord b => Ord (Dual b) Show basis => Show (Dual basis)