GenericNonAssociativeAlgebra(R, n, ls, gamma)

generic.spad line 1 [edit on github]

• R: CommutativeRing

• n: PositiveInteger

• ls: List Symbol

• gamma: Vector Matrix R

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 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, 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

Hashable if Fraction Polynomial R has Hashable

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