AlgebraGivenByStructuralConstants(R, n, ls, gamma)

naalg.spad line 1 [edit on github]

• R : CommutativeRing
• n : PositiveInteger
• ls : List(Symbol)
• gamma : Vector(Matrix(R))

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.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

* : (R, %) -> %

from LeftModule(R)

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

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

from LeftModule(SquareMatrix(n, R))

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

= : (%, %) -> Boolean

from BasicType

^ : (%, PositiveInteger) -> %

from Magma

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 : 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.

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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 : Vector(R) -> %

from FramedModule(R)

convert : % -> InputForm if R has Finite

from ConvertibleTo(InputForm)

convert : % -> Vector(R)

from FramedModule(R)

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

from FramedModule(R)

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

from FiniteRankNonAssociativeAlgebra(R)

coordinates : % -> Vector(R)

from FramedModule(R)

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

from FiniteRankNonAssociativeAlgebra(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 : (Mapping(R, OrderedVariableList(ls)), %) -> 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 : (Mapping(R, R), %) -> %

from IndexedProductCategory(R, OrderedVariableList(ls))

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

from IndexedProductCategory(R, OrderedVariableList(ls))

monomial? : % -> Boolean

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) -> %

from FramedModule(R)

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

from FiniteRankNonAssociativeAlgebra(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

~= : (%, %) -> Boolean

from BasicType

Comparable

ConvertibleTo(InputForm)

FiniteRankNonAssociativeAlgebra(R)

AbelianMonoid

BiModule(R, R)

IndexedProductCategory(R, OrderedVariableList(ls))

NonAssociativeAlgebra(R)

CancellationAbelianMonoid

unitsKnown

AbelianGroup

Module(R)

LeftModule(R)

SetCategory

CoercibleTo(OutputForm)

FramedModule(R)

Magma

AbelianSemiGroup

FramedNonAssociativeAlgebra(R)

IndexedDirectProductCategory(R, OrderedVariableList(ls))

FreeModuleCategory(R, OrderedVariableList(ls))

LeftModule(SquareMatrix(n, R))

RightModule(R)

AbelianProductCategory(R)

NonAssociativeRng

NonAssociativeSemiRng

Hashable

Finite

BasicType