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.

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 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 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

Hashable if R has Hashable

LeftModule R

Magma

Module R

NonAssociativeAlgebra R

NonAssociativeRng

NonAssociativeSemiRng

RightModule R

SetCategory

unitsKnown if R has IntegralDomain