AssociatedLieAlgebra(R, A)ΒΆ

lie.spad line 1

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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(R, R)

CancellationAbelianMonoid

CoercibleTo A

CoercibleTo OutputForm

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

LeftModule R

Magma

Module R

NonAssociativeAlgebra R

NonAssociativeRng

NonAssociativeSemiRng

RightModule R

SetCategory

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