# AssociatedLieAlgebra(R, A)ΒΆ

lie.spad line 1 [edit on github]

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

- ^: (%, PositiveInteger) -> %
from Magma

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

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

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

- antiCommutator: (%, %) -> %

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

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

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

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

- 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

- 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

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

- 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 R has Finite and A has FramedNonAssociativeAlgebra R
from Finite

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

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

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

- latex: % -> String
from SetCategory

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

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

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

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

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

- leftNorm: % -> R if A has FiniteRankNonAssociativeAlgebra R

- leftPower: (%, PositiveInteger) -> %
from Magma

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

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

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

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

- leftTrace: % -> R if A has FiniteRankNonAssociativeAlgebra R

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

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

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

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

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

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

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

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

- opposite?: (%, %) -> Boolean
from AbelianMonoid

- plenaryPower: (%, PositiveInteger) -> %
from NonAssociativeAlgebra R

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

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

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

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

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

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

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

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

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

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

- rightNorm: % -> R if A has FiniteRankNonAssociativeAlgebra R

- rightPower: (%, PositiveInteger) -> %
from Magma

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

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

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

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

- rightTrace: % -> R if A has FiniteRankNonAssociativeAlgebra R

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

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

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

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

- 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

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

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

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

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

- 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

Hashable if A has FramedNonAssociativeAlgebra R and R has Hashable

Module R

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