FramedNonAssociativeAlgebra R

naalgc.spad line 897 [edit on github]

FramedNonAssociativeAlgebra(R) is a FiniteRankNonAssociativeAlgebra (i.e. a non associative algebra over R which is a free R-module of finite rank) over a commutative ring R together with a fixed R-module basis.

0: %

from AbelianMonoid

*: (%, %) -> %

from Magma

*: (%, R) -> %

from RightModule R

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule 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, %) -> %

apply(m, a) defines a left operation of n by n matrices where n is the rank of the algebra in terms of matrix-vector multiplication, this is a substitute for a left module structure. Error: if shape of matrix doesn't fit.

associative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

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

from NonAssociativeRng

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

from FiniteRankNonAssociativeAlgebra R

basis: () -> Vector %

from FramedModule R

coerce: % -> OutputForm

from CoercibleTo OutputForm

commutative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

commutator: (%, %) -> %

from NonAssociativeRng

conditionsForIdempotents: () -> List Polynomial R

conditionsForIdempotents() determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed R-module basis.

conditionsForIdempotents: Vector % -> List Polynomial R

from FiniteRankNonAssociativeAlgebra R

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

elt(a, i) returns the i-th coefficient of a with respect to the fixed R-module basis.

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

from Finite

flexible?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

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

leftAlternative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial R

from FiniteRankNonAssociativeAlgebra R

leftDiscriminant: () -> R

leftDiscriminant() returns the determinant of the n-by-n matrix whose element at the i-th row and j-th column is given by the left trace of the product vi*vj, where v1, …, vn are the elements of the fixed R-module basis. Note: the same as determinant(leftTraceMatrix()).

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

leftRankPolynomial() calculates the left minimal polynomial of the generic element in the algebra, defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.

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

from FiniteRankNonAssociativeAlgebra R

leftRegularRepresentation: % -> Matrix R

leftRegularRepresentation(a) returns the matrix of the linear map defined by left multiplication by a with respect to the fixed R-module basis.

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

from FiniteRankNonAssociativeAlgebra R

leftTrace: % -> R

from FiniteRankNonAssociativeAlgebra R

leftTraceMatrix: () -> Matrix R

leftTraceMatrix() is the n-by-n matrix whose element at the i-th row and j-th column is given by left trace of the product vi*vj, where v1, …, vn are the elements of the fixed R-module basis.

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

lookup: % -> PositiveInteger if R has Finite

from Finite

noncommutativeJordanAlgebra?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra R

powerAssociative?: () -> Boolean

from FiniteRankNonAssociativeAlgebra R

random: () -> % if R has Finite

from Finite

rank: () -> PositiveInteger

from FramedModule R

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

from FiniteRankNonAssociativeAlgebra R

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

rightDiscriminant() returns the determinant of the n-by-n matrix whose element at the i-th row and j-th column is given by the right trace of the product vi*vj, where v1, …, vn are the elements of the fixed R-module basis. Note: the same as determinant(rightTraceMatrix()).

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

rightRankPolynomial() calculates the right minimal polynomial of the generic element in the algebra, defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.

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

from FiniteRankNonAssociativeAlgebra R

rightRegularRepresentation: % -> Matrix R

rightRegularRepresentation(a) returns the matrix of the linear map defined by right multiplication by a with respect to the fixed R-module basis.

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

from FiniteRankNonAssociativeAlgebra R

rightTrace: % -> R

from FiniteRankNonAssociativeAlgebra R

rightTraceMatrix: () -> Matrix R

rightTraceMatrix() is the n-by-n matrix whose element at the i-th row and j-th column is given by the right trace of the product vi*vj, where v1, …, vn are the elements of the fixed R-module basis.

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 Finite

from Comparable

someBasis: () -> Vector %

from FiniteRankNonAssociativeAlgebra R

structuralConstants: () -> Vector Matrix R

structuralConstants() calculates the structural constants [(gammaijk) for k in 1..rank()] defined by vi * vj = gammaij1 * v1 + ... + gammaijn * vn, where v1, …, vn is the fixed R-module basis.

structuralConstants: Vector % -> Vector Matrix R

from FiniteRankNonAssociativeAlgebra R

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

from CancellationAbelianMonoid

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

from FiniteRankNonAssociativeAlgebra R

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo OutputForm

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

Finite if R has Finite

FiniteRankNonAssociativeAlgebra R

FramedModule R

LeftModule R

Magma

Module R

NonAssociativeAlgebra R

NonAssociativeRng

NonAssociativeSemiRng

RightModule R

SetCategory

unitsKnown if R has IntegralDomain