GenericNonAssociativeAlgebra(R, n, ls, gamma)
generic.spad line 1
[edit on github]
AlgebraGenericElementPackage allows you to create generic elements of an algebra, i.e. the scalars are extended to include symbolic coefficients
- * : (%, %) -> %
- 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
- 0 : () -> %
- from AbelianMonoid
- = : (%, %) -> Boolean
- from BasicType
- ^ : (%, PositiveInteger) -> %
- from Magma
- 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 : 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
- coerce : % -> OutputForm
- from CoercibleTo(OutputForm)
- commutative? : () -> Boolean
- from FiniteRankNonAssociativeAlgebra(Fraction(Polynomial(R)))
- commutator : (%, %) -> %
- from NonAssociativeRng
- 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(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
- conditionsForIdempotents : () -> List(Polynomial(Fraction(Polynomial(R))))
- from FramedNonAssociativeAlgebra(Fraction(Polynomial(R)))
- conditionsForIdempotents : Vector(%) -> List(Polynomial(Fraction(Polynomial(R))))
- from FiniteRankNonAssociativeAlgebra(Fraction(Polynomial(R)))
- convert : Vector(Fraction(Polynomial(R))) -> %
- from FramedModule(Fraction(Polynomial(R)))
- convert : % -> InputForm if Fraction(Polynomial(R)) has Finite
- from ConvertibleTo(InputForm)
- convert : % -> Vector(Fraction(Polynomial(R)))
- from FramedModule(Fraction(Polynomial(R)))
- coordinates : Vector(%) -> Matrix(Fraction(Polynomial(R)))
- from FramedModule(Fraction(Polynomial(R)))
- coordinates : (Vector(%), Vector(%)) -> Matrix(Fraction(Polynomial(R)))
- from FiniteRankNonAssociativeAlgebra(Fraction(Polynomial(R)))
- coordinates : % -> Vector(Fraction(Polynomial(R)))
- from FramedModule(Fraction(Polynomial(R)))
- coordinates : (%, Vector(%)) -> Vector(Fraction(Polynomial(R)))
- from FiniteRankNonAssociativeAlgebra(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 -> %
generic(s) returns a generic element, i.e. the linear combination of the fixed basis with the symbolic coefficients s1, s2, ..
- 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(%) -> %
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
- 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
- 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 if Fraction(Polynomial(R)) has Hashable
- from Hashable
- hashUpdate! : (HashState, %) -> HashState if Fraction(Polynomial(R)) has Hashable
- from Hashable
- 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))) -> %
- from FramedModule(Fraction(Polynomial(R)))
- represents : (Vector(Fraction(Polynomial(R))), Vector(%)) -> %
- from FiniteRankNonAssociativeAlgebra(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
- ~= : (%, %) -> Boolean
- from BasicType
Comparable
ConvertibleTo(InputForm)
NonAssociativeSemiRng
LeftModule(Fraction(Polynomial(R)))
NonAssociativeAlgebra(Fraction(Polynomial(R)))
CancellationAbelianMonoid
AbelianGroup
FramedModule(Fraction(Polynomial(R)))
SetCategory
CoercibleTo(OutputForm)
RightModule(Fraction(Polynomial(R)))
Magma
unitsKnown
AbelianSemiGroup
FramedNonAssociativeAlgebra(Fraction(Polynomial(R)))
Module(Fraction(Polynomial(R)))
LeftModule(SquareMatrix(n, Fraction(Polynomial(R))))
BiModule(Fraction(Polynomial(R)), Fraction(Polynomial(R)))
NonAssociativeRng
FiniteRankNonAssociativeAlgebra(Fraction(Polynomial(R)))
AbelianMonoid
Hashable
Finite
BasicType