# LieSquareMatrix(n, R)ΒΆ

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
antiAssociative?: () -> Boolean
antiCommutative?: () -> Boolean
antiCommutator: (%, %) -> %
apply: (Matrix R, %) -> %
associative?: () -> Boolean
associator: (%, %, %) -> %
associatorDependence: () -> List Vector R if R has IntegralDomain
basis: () -> Vector %

from FramedModule R

coerce: % -> OutputForm
coerce: % -> SquareMatrix(n, R)

from CoercibleTo SquareMatrix(n, R)

commutative?: () -> Boolean
commutator: (%, %) -> %
conditionsForIdempotents: () -> List Polynomial R
conditionsForIdempotents: Vector % -> List Polynomial R
convert: % -> InputForm if R has Finite
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
coordinates: (Vector %, Vector %) -> Matrix R
coordinates: Vector % -> Matrix R

from FramedModule R

elt: (%, Integer) -> R
enumerate: () -> List % if R has Finite

from Finite

flexible?: () -> Boolean
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
jordanAlgebra?: () -> Boolean
latex: % -> String

from SetCategory

leftAlternative?: () -> Boolean
leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
leftDiscriminant: () -> R
leftDiscriminant: Vector % -> R
leftMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain
leftNorm: % -> R
leftPower: (%, PositiveInteger) -> %

from Magma

leftRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if R has Field
leftRecip: % -> Union(%, failed) if R has IntegralDomain
leftRegularRepresentation: % -> Matrix R
leftRegularRepresentation: (%, Vector %) -> Matrix R
leftTrace: % -> R
leftTraceMatrix: () -> Matrix R
leftTraceMatrix: Vector % -> Matrix R
leftUnit: () -> Union(%, failed) if R has IntegralDomain
leftUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain
lieAlgebra?: () -> Boolean
lookup: % -> PositiveInteger if R has Finite

from Finite

noncommutativeJordanAlgebra?: () -> Boolean
opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %
powerAssociative?: () -> Boolean
random: () -> % if R has Finite

from Finite

rank: () -> PositiveInteger

from FramedModule R

recip: % -> Union(%, failed) if R has IntegralDomain
represents: (Vector R, Vector %) -> %
represents: Vector R -> %

from FramedModule R

rightAlternative?: () -> Boolean
rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
rightDiscriminant: () -> R
rightDiscriminant: Vector % -> R
rightMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain
rightNorm: % -> R
rightPower: (%, PositiveInteger) -> %

from Magma

rightRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if R has Field
rightRecip: % -> Union(%, failed) if R has IntegralDomain
rightRegularRepresentation: % -> Matrix R
rightRegularRepresentation: (%, Vector %) -> Matrix R
rightTrace: % -> R
rightTraceMatrix: () -> Matrix R
rightTraceMatrix: Vector % -> Matrix R
rightUnit: () -> Union(%, failed) if R has IntegralDomain
rightUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain
sample: %

from AbelianMonoid

size: () -> NonNegativeInteger if R has Finite

from Finite

smaller?: (%, %) -> Boolean if R has Finite

from Comparable

someBasis: () -> Vector %
structuralConstants: () -> Vector Matrix R
structuralConstants: Vector % -> Vector Matrix R
subtractIfCan: (%, %) -> Union(%, failed)
unit: () -> Union(%, failed) if R has IntegralDomain
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo SquareMatrix(n, R)

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

Finite if R has Finite

Hashable if R has Hashable

Magma

NonAssociativeRng

NonAssociativeSemiRng

SetCategory

unitsKnown if R has IntegralDomain