FramedNonAssociativeAlgebra RΒΆ

naalgc.spad line 890

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