GenericNonAssociativeAlgebra(R, n, ls, gamma)ΒΆ

generic.spad line 1

AlgebraGenericElementPackage allows you to create generic elements of an algebra, i.e. the scalars are extended to include symbolic coefficients

0: %
from AbelianMonoid
*: (%, %) -> %
from Magma
*: (%, Fraction Polynomial R) -> %
from RightModule Fraction Polynomial R
*: (Fraction Polynomial R, %) -> %
from LeftModule Fraction Polynomial R
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (SquareMatrix(n, Fraction Polynomial R), %) -> %
from LeftModule SquareMatrix(n, Fraction Polynomial R)
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
=: (%, %) -> Boolean
from BasicType
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
alternative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
antiAssociative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
antiCommutative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
apply: (Matrix Fraction Polynomial R, %) -> %
from FramedNonAssociativeAlgebra Fraction Polynomial R
associative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
associator: (%, %, %) -> %
from NonAssociativeRng
associatorDependence: () -> List Vector Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
basis: () -> Vector %
from FramedModule Fraction Polynomial R
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Vector Fraction Polynomial R -> %
coerce(v) assumes that it is called with a vector of length equal to the dimension of the algebra, then a linear combination with the basis element is formed
commutative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
commutator: (%, %) -> %
from NonAssociativeRng
conditionsForIdempotents: () -> List Polynomial Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
conditionsForIdempotents: () -> List Polynomial R if R has IntegralDomain
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 Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
conditionsForIdempotents: Vector % -> List Polynomial R if R has IntegralDomain
conditionsForIdempotents([v1, ..., vn]) determines a complete list of polynomial equations for the coefficients of idempotents with respect to the R-module basis v1, ..., vn
convert: % -> InputForm if Fraction Polynomial R has Finite
from ConvertibleTo InputForm
convert: % -> Vector Fraction Polynomial R
from FramedModule Fraction Polynomial R
convert: Vector Fraction Polynomial R -> %
from FramedModule Fraction Polynomial R
coordinates: % -> Vector Fraction Polynomial R
from FramedModule Fraction Polynomial R
coordinates: (%, Vector %) -> Vector Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
coordinates: (Vector %, Vector %) -> Matrix Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
coordinates: Vector % -> Matrix Fraction Polynomial R
from FramedModule Fraction Polynomial R
elt: (%, Integer) -> Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
enumerate: () -> List % if Fraction Polynomial R has Finite
from Finite
flexible?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
generic: () -> %
generic() returns a generic element, i.e. the linear combination of the fixed basis with the symbolic coefficients \%x1, \%x2, ..
generic: (Symbol, Vector %) -> %
generic(s, v) returns a generic element, i.e. the linear combination of v with the symbolic coefficients s1, s2, ..
generic: (Vector Symbol, Vector %) -> %
generic(vs, ve) returns a generic element, i.e. the linear combination of ve with the symbolic coefficients vs error, if the vector of symbols is shorter than the vector of elements
generic: Symbol -> %
generic(s) returns a generic element, i.e. the linear combination of the fixed basis with the symbolic coefficients s1, s2, ..
generic: Vector % -> %
generic(ve) returns a generic element, i.e. the linear combination of ve basis with the symbolic coefficients \%x1, \%x2, ..
generic: Vector Symbol -> %
generic(vs) returns a generic element, i.e. the linear combination of the fixed basis with the symbolic coefficients vs; error, if the vector of symbols is too short
genericLeftDiscriminant: () -> Fraction Polynomial R if R has IntegralDomain
genericLeftDiscriminant() is the determinant of the generic left trace forms of all products of basis element, if the generic left trace form is associative, an algebra is separable if the generic left discriminant is invertible, if it is non-zero, there is some ring extension which makes the algebra separable
genericLeftMinimalPolynomial: % -> SparseUnivariatePolynomial Fraction Polynomial R if R has IntegralDomain
genericLeftMinimalPolynomial(a) substitutes the coefficients of {em a} for the generic coefficients in leftRankPolynomial()
genericLeftNorm: % -> Fraction Polynomial R if R has IntegralDomain
genericLeftNorm(a) substitutes the coefficients of a for the generic coefficients into the coefficient of the constant term in leftRankPolynomial and changes the sign if the degree of this polynomial is odd. This is a form of degree k
genericLeftTrace: % -> Fraction Polynomial R if R has IntegralDomain
genericLeftTrace(a) substitutes the coefficients of a for the generic coefficients into the coefficient of the second highest term in leftRankPolynomial and changes the sign. This is a linear form
genericLeftTraceForm: (%, %) -> Fraction Polynomial R if R has IntegralDomain
genericLeftTraceForm (a, b) is defined to be genericLeftTrace (a*b), this defines a symmetric bilinear form on the algebra
genericRightDiscriminant: () -> Fraction Polynomial R if R has IntegralDomain
genericRightDiscriminant() is the determinant of the generic left trace forms of all products of basis element, if the generic left trace form is associative, an algebra is separable if the generic left discriminant is invertible, if it is non-zero, there is some ring extension which makes the algebra separable
genericRightMinimalPolynomial: % -> SparseUnivariatePolynomial Fraction Polynomial R if R has IntegralDomain
genericRightMinimalPolynomial(a) substitutes the coefficients of a for the generic coefficients in rightRankPolynomial
genericRightNorm: % -> Fraction Polynomial R if R has IntegralDomain
genericRightNorm(a) substitutes the coefficients of a for the generic coefficients into the coefficient of the constant term in rightRankPolynomial and changes the sign if the degree of this polynomial is odd
genericRightTrace: % -> Fraction Polynomial R if R has IntegralDomain
genericRightTrace(a) substitutes the coefficients of a for the generic coefficients into the coefficient of the second highest term in rightRankPolynomial and changes the sign
genericRightTraceForm: (%, %) -> Fraction Polynomial R if R has IntegralDomain
genericRightTraceForm (a, b) is defined to be genericRightTrace (a*b), this defines a symmetric bilinear form on the algebra
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
index: PositiveInteger -> % if Fraction Polynomial R has Finite
from Finite
jacobiIdentity?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
jordanAdmissible?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
jordanAlgebra?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
latex: % -> String
from SetCategory
leftAlternative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftDiscriminant: () -> Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
leftDiscriminant: Vector % -> Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftMinimalPolynomial: % -> SparseUnivariatePolynomial Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftNorm: % -> Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftPower: (%, PositiveInteger) -> %
from Magma
leftRankPolynomial: () -> SparseUnivariatePolynomial Fraction Polynomial R if R has IntegralDomain
leftRankPolynomial() returns the left minimimal polynomial of the generic element
leftRankPolynomial: () -> SparseUnivariatePolynomial Polynomial Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
leftRecip: % -> Union(%, failed)
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftRegularRepresentation: % -> Matrix Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
leftRegularRepresentation: (%, Vector %) -> Matrix Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftTrace: % -> Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftTraceMatrix: () -> Matrix Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
leftTraceMatrix: Vector % -> Matrix Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftUnit: () -> Union(%, failed)
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
leftUnits: () -> Union(Record(particular: %, basis: List %), failed)
leftUnits() returns the affine space of all left units of the algebra, or "failed" if there is none
lieAdmissible?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
lieAlgebra?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
lookup: % -> PositiveInteger if Fraction Polynomial R has Finite
from Finite
noncommutativeJordanAlgebra?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
opposite?: (%, %) -> Boolean
from AbelianMonoid
plenaryPower: (%, PositiveInteger) -> %
from NonAssociativeAlgebra Fraction Polynomial R
powerAssociative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
random: () -> % if Fraction Polynomial R has Finite
from Finite
rank: () -> PositiveInteger
from FramedModule Fraction Polynomial R
recip: % -> Union(%, failed)
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
represents: (Vector Fraction Polynomial R, Vector %) -> %
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
represents: Vector Fraction Polynomial R -> %
from FramedModule Fraction Polynomial R
rightAlternative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightDiscriminant: () -> Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
rightDiscriminant: Vector % -> Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightMinimalPolynomial: % -> SparseUnivariatePolynomial Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightNorm: % -> Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightPower: (%, PositiveInteger) -> %
from Magma
rightRankPolynomial: () -> SparseUnivariatePolynomial Fraction Polynomial R if R has IntegralDomain
rightRankPolynomial() returns the right minimimal polynomial of the generic element
rightRankPolynomial: () -> SparseUnivariatePolynomial Polynomial Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
rightRecip: % -> Union(%, failed)
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightRegularRepresentation: % -> Matrix Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
rightRegularRepresentation: (%, Vector %) -> Matrix Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightTrace: % -> Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightTraceMatrix: () -> Matrix Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
rightTraceMatrix: Vector % -> Matrix Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightUnit: () -> Union(%, failed)
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
rightUnits: () -> Union(Record(particular: %, basis: List %), failed)
rightUnits() returns the affine space of all right units of the algebra, or "failed" if there is none
sample: %
from AbelianMonoid
size: () -> NonNegativeInteger if Fraction Polynomial R has Finite
from Finite
smaller?: (%, %) -> Boolean if Fraction Polynomial R has Finite
from Comparable
someBasis: () -> Vector %
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
structuralConstants: () -> Vector Matrix Fraction Polynomial R
from FramedNonAssociativeAlgebra Fraction Polynomial R
structuralConstants: Vector % -> Vector Matrix Fraction Polynomial R
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
unit: () -> Union(%, failed)
from FiniteRankNonAssociativeAlgebra Fraction Polynomial R
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(Fraction Polynomial R, Fraction Polynomial R)

CancellationAbelianMonoid

CoercibleTo OutputForm

Comparable if Fraction Polynomial R has Finite

ConvertibleTo InputForm if Fraction Polynomial R has Finite

Finite if Fraction Polynomial R has Finite

FiniteRankNonAssociativeAlgebra Fraction Polynomial R

FramedModule Fraction Polynomial R

FramedNonAssociativeAlgebra Fraction Polynomial R

LeftModule Fraction Polynomial R

LeftModule SquareMatrix(n, Fraction Polynomial R)

Magma

Module Fraction Polynomial R

NonAssociativeAlgebra Fraction Polynomial R

NonAssociativeRng

NonAssociativeSemiRng

RightModule Fraction Polynomial R

SetCategory

unitsKnown