LieSquareMatrix(n, R)ΒΆ

lie.spad line 109

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.

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, %) -> %
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: % -> InputForm if R has Finite
from ConvertibleTo InputForm
convert: % -> Vector R
from FramedModule R
convert: SquareMatrix(n, R) -> %
converts a SquareMatrix to a LieSquareMatrix
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
from FramedNonAssociativeAlgebra R
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
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 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
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo OutputForm

CoercibleTo SquareMatrix(n, R)

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

Finite if R has Finite

FiniteRankNonAssociativeAlgebra R

FramedModule R

FramedNonAssociativeAlgebra R

LeftModule R

Magma

Module R

NonAssociativeAlgebra R

NonAssociativeRng

NonAssociativeSemiRng

RightModule R

SetCategory

unitsKnown if R has IntegralDomain