# 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
- 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

BiModule(R, R)

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

FiniteRankNonAssociativeAlgebra R

Module R

unitsKnown if R has IntegralDomain