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.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

* : (R, %) -> %

from LeftModule(R)

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

= : (%, %) -> Boolean

from BasicType

^ : (%, PositiveInteger) -> %

from Magma

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 -> %

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

coerce : % -> A

from CoercibleTo(A)

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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

from FramedModule(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)

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

from FramedModule(R)

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

from FiniteRankNonAssociativeAlgebra(R)

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

from FramedModule(R)

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

from FiniteRankNonAssociativeAlgebra(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) -> % if A has FramedNonAssociativeAlgebra(R)

from FramedModule(R)

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

from FiniteRankNonAssociativeAlgebra(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

~= : (%, %) -> Boolean

from BasicType

Comparable

ConvertibleTo(InputForm)

FiniteRankNonAssociativeAlgebra(R)

AbelianMonoid

BiModule(R, R)

NonAssociativeAlgebra(R)

CancellationAbelianMonoid

unitsKnown

AbelianGroup

LeftModule(R)

SetCategory

FramedModule(R)

Magma

CoercibleTo(A)

CoercibleTo(OutputForm)

AbelianSemiGroup

FramedNonAssociativeAlgebra(R)

Module(R)

RightModule(R)

NonAssociativeRng

NonAssociativeSemiRng

Hashable

Finite

BasicType