FramedNonAssociativeAlgebra(R)

naalgc.spad line 895 [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.

* : (%, %) -> %
from Magma
* : (%, R) -> %
from RightModule(R)
* : (R, %) -> %
from LeftModule(R)
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
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), %) -> %

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 : 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

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

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
~= : (%, %) -> Boolean
from BasicType

Comparable

ConvertibleTo(InputForm)

NonAssociativeSemiRng

FiniteRankNonAssociativeAlgebra(R)

AbelianMonoid

BiModule(R, R)

NonAssociativeAlgebra(R)

LeftModule(R)

SetCategory

CoercibleTo(OutputForm)

FramedModule(R)

Magma

NonAssociativeRng

unitsKnown

AbelianGroup

AbelianSemiGroup

CancellationAbelianMonoid

RightModule(R)

Module(R)

Hashable

Finite

BasicType