LieSquareMatrix(n, R)

lie.spad line 109 [edit on github]

• n : PositiveInteger
• R : CommutativeRing

LieSquareMatrix(n, R) implements the Lie algebra of the n by n matrices over the commutative ring R. The Lie bracket (commutator) of the algebra is given by a*b := (a *$SQMATRIX(n, R) b - b *$SQMATRIX(n, R) a), where *$SQMATRIX(n, R) is the usual matrix multiplication.

* : (%, %) -> %

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), %) -> %

from FramedNonAssociativeAlgebra(R)

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)

coerce : % -> SquareMatrix(n, R)

from CoercibleTo(SquareMatrix(n, R))

commutative? : () -> Boolean

from FiniteRankNonAssociativeAlgebra(R)

commutator : (%, %) -> %

from NonAssociativeRng

conditionsForIdempotents : () -> List(Polynomial(R))

from FramedNonAssociativeAlgebra(R)

conditionsForIdempotents : Vector(%) -> List(Polynomial(R))

from FiniteRankNonAssociativeAlgebra(R)

convert : SquareMatrix(n, R) -> %

converts a SquareMatrix to a LieSquareMatrix

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

from FramedNonAssociativeAlgebra(R)

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

from FramedNonAssociativeAlgebra(R)

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

from FramedNonAssociativeAlgebra(R)

leftRecip : % -> Union(%, "failed") if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra(R)

leftRegularRepresentation : % -> Matrix(R)

from FramedNonAssociativeAlgebra(R)

leftRegularRepresentation : (%, Vector(%)) -> Matrix(R)

from FiniteRankNonAssociativeAlgebra(R)

leftTrace : % -> R

from FiniteRankNonAssociativeAlgebra(R)

leftTraceMatrix : () -> Matrix(R)

from FramedNonAssociativeAlgebra(R)

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

from FramedNonAssociativeAlgebra(R)

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

from FramedNonAssociativeAlgebra(R)

rightRecip : % -> Union(%, "failed") if R has IntegralDomain

from FiniteRankNonAssociativeAlgebra(R)

rightRegularRepresentation : % -> Matrix(R)

from FramedNonAssociativeAlgebra(R)

rightRegularRepresentation : (%, Vector(%)) -> Matrix(R)

from FiniteRankNonAssociativeAlgebra(R)

rightTrace : % -> R

from FiniteRankNonAssociativeAlgebra(R)

rightTraceMatrix : () -> Matrix(R)

from FramedNonAssociativeAlgebra(R)

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))

from FramedNonAssociativeAlgebra(R)

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)

FiniteRankNonAssociativeAlgebra(R)

AbelianMonoid

BiModule(R, R)

NonAssociativeAlgebra(R)

AbelianGroup

Module(R)

LeftModule(R)

SetCategory

CoercibleTo(OutputForm)

FramedModule(R)

Magma

unitsKnown

AbelianSemiGroup

FramedNonAssociativeAlgebra(R)

CoercibleTo(SquareMatrix(n, R))

CancellationAbelianMonoid

RightModule(R)

NonAssociativeRng

NonAssociativeSemiRng

Hashable

Finite

BasicType