AssociatedLieAlgebra(R, A)

lie.spad line 1 [edit on github]

• R: CommutativeRing

• A: NonAssociativeAlgebra R

AssociatedLieAlgebra takes an algebra A and uses *$A to define the Lie bracket a*b := (a *\$A b - b *\$A a) (commutator). Note that the notation [a, b] cannot be used due to restrictions of the current compiler. This domain only gives a Lie algebra if the Jacobi-identity (a*b)*c + (b*c)*a + (c*a)*b = 0 holds for all a, b, c in A. This relation can be checked by lieAdmissible?()\$A. If the underlying algebra is of type FramedNonAssociativeAlgebra(R) (i.e. a non associative algebra over R which is a free R-module of finite rank, together with a fixed R-module basis), then the same is true for the associated Lie algebra. Also, if the underlying algebra is of type FiniteRankNonAssociativeAlgebra(R) (i.e. a non associative algebra over R which is a free R-module of finite rank), then the same is true for the associated Lie algebra.

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 if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

antiAssociative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

antiCommutative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

apply: (Matrix R, %) -> % if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

associative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

associator: (%, %, %) -> %

from NonAssociativeRng

associatorDependence: () -> List Vector R if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

basis: () -> Vector % if A has FramedNonAssociativeAlgebra R

from FramedModule R

coerce: % -> A

from CoercibleTo A

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: A -> %

coerce(a) coerces the element a of the algebra A to an element of the Lie algebra AssociatedLieAlgebra(R, A).

commutative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

commutator: (%, %) -> %

from NonAssociativeRng

conditionsForIdempotents: () -> List Polynomial R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

conditionsForIdempotents: Vector % -> List Polynomial R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

convert: % -> InputForm if A has FramedNonAssociativeAlgebra R and R has Finite

from ConvertibleTo InputForm

convert: % -> Vector R if A has FramedNonAssociativeAlgebra R

from FramedModule R

convert: Vector R -> % if A has FramedNonAssociativeAlgebra R

from FramedModule R

coordinates: % -> Vector R if A has FramedNonAssociativeAlgebra R

from FramedModule R

coordinates: (%, Vector %) -> Vector R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

coordinates: (Vector %, Vector %) -> Matrix R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

coordinates: Vector % -> Matrix R if A has FramedNonAssociativeAlgebra R

from FramedModule R

elt: (%, Integer) -> R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

enumerate: () -> List % if A has FramedNonAssociativeAlgebra R and R has Finite

from Finite

flexible?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

hash: % -> SingleInteger if A has FramedNonAssociativeAlgebra R and R has Hashable

from Hashable

hashUpdate!: (HashState, %) -> HashState if A has FramedNonAssociativeAlgebra R and R has Hashable

from Hashable

index: PositiveInteger -> % if A has FramedNonAssociativeAlgebra R and R has Finite

from Finite

jacobiIdentity?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

jordanAdmissible?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

jordanAlgebra?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

latex: % -> String

from SetCategory

leftAlternative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftDiscriminant: () -> R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

leftDiscriminant: Vector % -> R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftMinimalPolynomial: % -> SparseUnivariatePolynomial R if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftNorm: % -> R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftPower: (%, PositiveInteger) -> %

from Magma

leftRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if A has FramedNonAssociativeAlgebra R and R has Field

from FramedNonAssociativeAlgebra R

leftRecip: % -> Union(%, failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftRegularRepresentation: % -> Matrix R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

leftRegularRepresentation: (%, Vector %) -> Matrix R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftTrace: % -> R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftTraceMatrix: () -> Matrix R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

leftTraceMatrix: Vector % -> Matrix R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftUnit: () -> Union(%, failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

leftUnits: () -> Union(Record(particular: %, basis: List %), failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

lieAdmissible?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

lieAlgebra?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

lookup: % -> PositiveInteger if A has FramedNonAssociativeAlgebra R and R has Finite

from Finite

noncommutativeJordanAlgebra?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra R

powerAssociative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

random: () -> % if A has FramedNonAssociativeAlgebra R and R has Finite

from Finite

rank: () -> PositiveInteger if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

recip: % -> Union(%, failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

represents: (Vector R, Vector %) -> % if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

represents: Vector R -> % if A has FramedNonAssociativeAlgebra R

from FramedModule R

rightAlternative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightDiscriminant: () -> R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

rightDiscriminant: Vector % -> R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightMinimalPolynomial: % -> SparseUnivariatePolynomial R if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightNorm: % -> R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightPower: (%, PositiveInteger) -> %

from Magma

rightRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if A has FramedNonAssociativeAlgebra R and R has Field

from FramedNonAssociativeAlgebra R

rightRecip: % -> Union(%, failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightRegularRepresentation: % -> Matrix R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

rightRegularRepresentation: (%, Vector %) -> Matrix R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightTrace: % -> R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightTraceMatrix: () -> Matrix R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

rightTraceMatrix: Vector % -> Matrix R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightUnit: () -> Union(%, failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

rightUnits: () -> Union(Record(particular: %, basis: List %), failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

sample: %

from AbelianMonoid

size: () -> NonNegativeInteger if A has FramedNonAssociativeAlgebra R and R has Finite

from Finite

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

from Comparable

someBasis: () -> Vector % if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

structuralConstants: () -> Vector Matrix R if A has FramedNonAssociativeAlgebra R

from FramedNonAssociativeAlgebra R

structuralConstants: Vector % -> Vector Matrix R if A has FiniteRankNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

unit: () -> Union(%, failed) if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R

from FiniteRankNonAssociativeAlgebra R

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo A

CoercibleTo OutputForm

Comparable if A has FramedNonAssociativeAlgebra R and R has Finite

ConvertibleTo InputForm if A has FramedNonAssociativeAlgebra R and R has Finite

Finite if A has FramedNonAssociativeAlgebra R and R has Finite

FiniteRankNonAssociativeAlgebra R if A has FiniteRankNonAssociativeAlgebra R

FramedModule R if A has FramedNonAssociativeAlgebra R

FramedNonAssociativeAlgebra R if A has FramedNonAssociativeAlgebra R

Hashable if A has FramedNonAssociativeAlgebra R and R has Hashable

LeftModule R

Magma

Module R

NonAssociativeAlgebra R

NonAssociativeRng

NonAssociativeSemiRng

RightModule R

SetCategory

unitsKnown if A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain or R has IntegralDomain and A has FramedNonAssociativeAlgebra R