# FramedNonAssociativeAlgebra R¶

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
antiAssociative?: () -> Boolean
antiCommutative?: () -> Boolean
antiCommutator: (%, %) -> %
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
associator: (%, %, %) -> %
associatorDependence: () -> List Vector R if R has IntegralDomain
basis: () -> Vector %

from FramedModule R

coerce: % -> OutputForm
commutative?: () -> Boolean
commutator: (%, %) -> %
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
convert: % -> InputForm if R has Finite
convert: % -> Vector R

from FramedModule R

convert: Vector R -> %

from FramedModule R

coordinates: % -> Vector R

from FramedModule R

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

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

index: PositiveInteger -> % if R has Finite

from Finite

jacobiIdentity?: () -> Boolean
jordanAlgebra?: () -> Boolean
latex: % -> String

from SetCategory

leftAlternative?: () -> Boolean
leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial 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
leftMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain
leftNorm: % -> 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
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
leftTrace: % -> 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
leftUnit: () -> Union(%, failed) if R has IntegralDomain
leftUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain
lieAlgebra?: () -> Boolean
lookup: % -> PositiveInteger if R has Finite

from Finite

noncommutativeJordanAlgebra?: () -> Boolean
opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %
powerAssociative?: () -> Boolean
random: () -> % if R has Finite

from Finite

rank: () -> PositiveInteger

from FramedModule R

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

from FramedModule R

rightAlternative?: () -> Boolean
rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial 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
rightMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain
rightNorm: % -> 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
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
rightTrace: % -> 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
rightUnit: () -> Union(%, failed) if R has IntegralDomain
rightUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain
sample: %

from AbelianMonoid

size: () -> NonNegativeInteger if R has Finite

from Finite

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

from Comparable

someBasis: () -> Vector %
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
subtractIfCan: (%, %) -> Union(%, failed)
unit: () -> Union(%, failed) if R has IntegralDomain
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

Finite if R has Finite

Magma

NonAssociativeRng

NonAssociativeSemiRng

SetCategory

unitsKnown if R has IntegralDomain