AlgebraGivenByStructuralConstants(R, n, ls, gamma)ΒΆ

naalg.spad line 1 [edit on github]

AlgebraGivenByStructuralConstants implements finite rank algebras over a commutative ring, given by the structural constants gamma with respect to a fixed basis [a1, .., an], where gamma is an n-vector of n by n matrices [(gammaijk) for k in 1..rank()] defined by ai * aj = gammaij1 * a1 + ... + gammaijn * an. The symbols for the fixed basis have to be given as a list of symbols.

0: %

from AbelianMonoid

*: (%, %) -> %

from Magma

*: (%, R) -> %

from RightModule R

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

*: (SquareMatrix(n, R), %) -> %

from LeftModule SquareMatrix(n, R)

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

alternative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

antiAssociative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

antiCommutative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

apply: (Matrix R, %) -> %

from FramedNonAssociativeAlgebra R

associative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

associator: (%, %, %) -> %

from NonAssociativeRng

associatorDependence: () -> List Vector R if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

basis: () -> Vector %

from FramedModule R

coefficient: (%, OrderedVariableList ls) -> R

from FreeModuleCategory(R, OrderedVariableList ls)

coefficients: % -> List R

from FreeModuleCategory(R, OrderedVariableList ls)

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Vector R -> %

coerce(v) converts a vector to a member of the algebra by forming a linear combination with the basis element. Note: the vector is assumed to have length equal to the dimension of the algebra.

commutative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

commutator: (%, %) -> %

from NonAssociativeRng

conditionsForIdempotents: () -> List Polynomial R

from FramedNonAssociativeAlgebra R

conditionsForIdempotents: Vector % -> List Polynomial R

from FiniteRankNonAssociativeAlgebra R

construct: List Record(k: OrderedVariableList ls, c: R) -> %

from IndexedProductCategory(R, OrderedVariableList ls)

constructOrdered: List Record(k: OrderedVariableList ls, c: R) -> %

from IndexedProductCategory(R, OrderedVariableList ls)

convert: % -> InputForm if R has Finite

from ConvertibleTo InputForm

convert: % -> Vector R

from FramedModule R

convert: Vector R -> %

from FramedModule R

coordinates: % -> Vector R

from FramedModule R

coordinates: (%, Vector %) -> Vector R

from FiniteRankNonAssociativeAlgebra R

coordinates: (Vector %, Vector %) -> Matrix R

from FiniteRankNonAssociativeAlgebra R

coordinates: Vector % -> Matrix R

from FramedModule R

elt: (%, Integer) -> R

from FramedNonAssociativeAlgebra R

enumerate: () -> List % if R has Finite

from Finite

flexible?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

hash: % -> SingleInteger if R has Hashable

from Hashable

hashUpdate!: (HashState, %) -> HashState if R has Hashable

from Hashable

index: PositiveInteger -> % if R has Finite

from Finite

jacobiIdentity?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

jordanAdmissible?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

jordanAlgebra?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

latex: % -> String

from SetCategory

leadingCoefficient: % -> R

from IndexedProductCategory(R, OrderedVariableList ls)

leadingMonomial: % -> %

from IndexedProductCategory(R, OrderedVariableList ls)

leadingSupport: % -> OrderedVariableList ls

from IndexedProductCategory(R, OrderedVariableList ls)

leadingTerm: % -> Record(k: OrderedVariableList ls, c: R)

from IndexedProductCategory(R, OrderedVariableList ls)

leftAlternative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial R

from FiniteRankNonAssociativeAlgebra R

leftDiscriminant: () -> R

from FramedNonAssociativeAlgebra R

leftDiscriminant: Vector % -> R

from FiniteRankNonAssociativeAlgebra R

leftMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

leftNorm: % -> R

from FiniteRankNonAssociativeAlgebra R

leftPower: (%, PositiveInteger) -> %

from Magma

leftRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if R has Field

from FramedNonAssociativeAlgebra R

leftRecip: % -> Union(%, failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

leftRegularRepresentation: % -> Matrix R

from FramedNonAssociativeAlgebra R

leftRegularRepresentation: (%, Vector %) -> Matrix R

from FiniteRankNonAssociativeAlgebra R

leftTrace: % -> R

from FiniteRankNonAssociativeAlgebra R

leftTraceMatrix: () -> Matrix R

from FramedNonAssociativeAlgebra R

leftTraceMatrix: Vector % -> Matrix R

from FiniteRankNonAssociativeAlgebra R

leftUnit: () -> Union(%, failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

leftUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

lieAdmissible?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

lieAlgebra?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

linearExtend: (OrderedVariableList ls -> R, %) -> R

from FreeModuleCategory(R, OrderedVariableList ls)

listOfTerms: % -> List Record(k: OrderedVariableList ls, c: R)

from IndexedDirectProductCategory(R, OrderedVariableList ls)

lookup: % -> PositiveInteger if R has Finite

from Finite

map: (R -> R, %) -> %

from IndexedProductCategory(R, OrderedVariableList ls)

monomial?: % -> Boolean

from IndexedProductCategory(R, OrderedVariableList ls)

monomial: (R, OrderedVariableList ls) -> %

from IndexedProductCategory(R, OrderedVariableList ls)

monomials: % -> List %

from FreeModuleCategory(R, OrderedVariableList ls)

noncommutativeJordanAlgebra?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(R, OrderedVariableList ls)

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra R

powerAssociative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

random: () -> % if R has Finite

from Finite

rank: () -> PositiveInteger

from FiniteRankNonAssociativeAlgebra R

recip: % -> Union(%, failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

reductum: % -> %

from IndexedProductCategory(R, OrderedVariableList ls)

represents: (Vector R, Vector %) -> %

from FiniteRankNonAssociativeAlgebra R

represents: Vector R -> %

from FramedModule R

rightAlternative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial R

from FiniteRankNonAssociativeAlgebra R

rightDiscriminant: () -> R

from FramedNonAssociativeAlgebra R

rightDiscriminant: Vector % -> R

from FiniteRankNonAssociativeAlgebra R

rightMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

rightNorm: % -> R

from FiniteRankNonAssociativeAlgebra R

rightPower: (%, PositiveInteger) -> %

from Magma

rightRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if R has Field

from FramedNonAssociativeAlgebra R

rightRecip: % -> Union(%, failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

rightRegularRepresentation: % -> Matrix R

from FramedNonAssociativeAlgebra R

rightRegularRepresentation: (%, Vector %) -> Matrix R

from FiniteRankNonAssociativeAlgebra R

rightTrace: % -> R

from FiniteRankNonAssociativeAlgebra R

rightTraceMatrix: () -> Matrix R

from FramedNonAssociativeAlgebra R

rightTraceMatrix: Vector % -> Matrix R

from FiniteRankNonAssociativeAlgebra R

rightUnit: () -> Union(%, failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

rightUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

sample: %

from AbelianMonoid

size: () -> NonNegativeInteger if R has Finite

from Finite

smaller?: (%, %) -> Boolean if R has Comparable

from Comparable

someBasis: () -> Vector %

from FiniteRankNonAssociativeAlgebra R

structuralConstants: () -> Vector Matrix R

from FramedNonAssociativeAlgebra R

structuralConstants: Vector % -> Vector Matrix R

from FiniteRankNonAssociativeAlgebra R

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

support: % -> List OrderedVariableList ls

from FreeModuleCategory(R, OrderedVariableList ls)

unit: () -> Union(%, failed) if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra R

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianProductCategory R

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo OutputForm

Comparable if R has Comparable

ConvertibleTo InputForm if R has Finite

Finite if R has Finite

FiniteRankNonAssociativeAlgebra R

FramedModule R

FramedNonAssociativeAlgebra R

FreeModuleCategory(R, OrderedVariableList ls)

Hashable if R has Hashable

IndexedDirectProductCategory(R, OrderedVariableList ls)

IndexedProductCategory(R, OrderedVariableList ls)

LeftModule R

LeftModule SquareMatrix(n, R)

Magma

Module R

NonAssociativeAlgebra R

NonAssociativeRng

NonAssociativeSemiRng

RightModule R

SetCategory

unitsKnown if R has IntegralDomain