AssociatedLieAlgebra(R, A)¶

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
antiAssociative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R
antiCommutative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R
antiCommutator: (%, %) -> %
apply: (Matrix R, %) -> % if A has FramedNonAssociativeAlgebra R
associative?: () -> Boolean if A has FiniteRankNonAssociativeAlgebra R
associator: (%, %, %) -> %
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
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: (%, %) -> %
conditionsForIdempotents: () -> List Polynomial R if A has FramedNonAssociativeAlgebra R
conditionsForIdempotents: Vector % -> List Polynomial R if A has FiniteRankNonAssociativeAlgebra R
convert: % -> InputForm if R has Finite and A has FramedNonAssociativeAlgebra R
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
enumerate: () -> List % if R has Finite and A has FramedNonAssociativeAlgebra R

from Finite

flexible?: () -> Boolean if A has 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
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
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
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
leftRegularRepresentation: (%, Vector %) -> Matrix R if A has FiniteRankNonAssociativeAlgebra R
leftTrace: % -> R if A has FiniteRankNonAssociativeAlgebra R
leftTraceMatrix: () -> Matrix R if A has 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) -> %
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
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
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
rightRegularRepresentation: (%, Vector %) -> Matrix R if A has FiniteRankNonAssociativeAlgebra R
rightTrace: % -> R if A has FiniteRankNonAssociativeAlgebra R
rightTraceMatrix: () -> Matrix R if A has 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
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

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

FramedModule R if A has FramedNonAssociativeAlgebra R

Magma

NonAssociativeRng

NonAssociativeSemiRng

SetCategory

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