# AssociatedLieAlgebra(R, A)ΒΆ

- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- 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 R has Finite and A has FramedNonAssociativeAlgebra R
- 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 R has Finite and A has FramedNonAssociativeAlgebra R
- from Finite
- flexible?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R
- from FiniteRankNonAssociativeAlgebra R
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- index: PositiveInteger -> % if R has Finite and A has FramedNonAssociativeAlgebra R
- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- from FiniteRankNonAssociativeAlgebra R
- leftUnits: () -> Union(Record(particular: %, basis: List %), failed) if A has FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- 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 R has Finite and A has FramedNonAssociativeAlgebra R
- 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 R has Finite and A has FramedNonAssociativeAlgebra R
- from Finite
- rank: () -> PositiveInteger if A has FiniteRankNonAssociativeAlgebra R
- from FiniteRankNonAssociativeAlgebra R
- recip: % -> Union(%, failed) if A has FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- from FiniteRankNonAssociativeAlgebra R
- rightUnits: () -> Union(Record(particular: %, basis: List %), failed) if A has FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- from FiniteRankNonAssociativeAlgebra R
- sample: %
- from AbelianMonoid
- size: () -> NonNegativeInteger if R has Finite and A has FramedNonAssociativeAlgebra R
- from Finite
- smaller?: (%, %) -> Boolean if R has Finite and A has FramedNonAssociativeAlgebra R
- 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 FramedNonAssociativeAlgebra R and R has IntegralDomain or A has FiniteRankNonAssociativeAlgebra R and R has IntegralDomain
- from FiniteRankNonAssociativeAlgebra R
- zero?: % -> Boolean
- from AbelianMonoid

BiModule(R, R)

Comparable if R has Finite and A has FramedNonAssociativeAlgebra R

ConvertibleTo InputForm if R has Finite and A has FramedNonAssociativeAlgebra R

Finite if R has Finite and A has FramedNonAssociativeAlgebra R

FiniteRankNonAssociativeAlgebra R if A has FiniteRankNonAssociativeAlgebra R

FramedModule R if A has FramedNonAssociativeAlgebra R

FramedNonAssociativeAlgebra R if A has FramedNonAssociativeAlgebra R

Module R

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