LieDerivation -- the class of all Lie algebra derivations

Description

Given a homomorphism of Lie algebras $f: M \ \to\ L$, one has the notion of a derivation $d: M \ \to\ L$ over $f$, and LieDerivation is the type representing such pairs $(d,\,f)$ ($f$ is the identity for the case of ordinary derivations from $L$ to $L$). The derivation law reads \break $d$ [x, y] = [$d$ x, $f$ y] ± [$f$ x, $d$ y], \break where the sign is determined by the sign of interchanging $d$ and $x$, i.e., the sign is plus if sign$(d)$=0 or sign$(x)$=0 and minus otherwise. An object of type LieDerivation need not be well defined as a map. Use isWellDefined(ZZ,LieDerivation) to check if the derivation is well defined.

i1 : L = lieAlgebra{a,b}

o1 = L

o1 : LieAlgebra

i2 : M = lieAlgebra{a,b,c}

o2 = M

o2 : LieAlgebra

i3 : f = map(L,M)

o3 = f

o3 : LieAlgebraMap

i4 : use L

i5 : der = lieDerivation(f,{a a b,b b a,a a b+b b a})

o5 = der

o5 : LieDerivation

i6 : describe der

o6 = a =>  - (a b a)
     b => (b b a)
     c =>  - (a b a) + (b b a)
     map => f
     sign => 0
     weight => {2, 0}
     source => M
     target => L

i7 : use M

i8 : der a c

o8 =  - (a a b a) + (b a b a)

o8 : L

Functions and methods returning an object of class LieDerivation :

euler(LieAlgebra) -- compute the Euler derivation
innerDerivation -- make the derivation defined by right Lie multiplication by a Lie element
lieDerivation -- make a graded derivation

Methods that use an object of class LieDerivation :

- LieDerivation -- unary negation
describe(LieDerivation) -- see describe(LieAlgebra) -- real description
firstDegree(LieDerivation) -- see firstDegree -- get the degree of an element
image(LieDerivation) -- make the image of a Lie derivation
image(LieDerivation,LieSubSpace) -- make the image of a Lie subspace under a Lie derivation
inverse(LieDerivation,LieSubSpace) -- make the inverse image of a Lie subspace under a Lie derivation
isWellDefined(ZZ,LieDerivation) -- whether a Lie derivation is well defined
kernel(LieDerivation) -- make the kernel of a map
LieAlgebraMap * LieDerivation -- composition of a homomorphism and a derivation
LieDerivation * LieAlgebraMap -- composition of a derivation and a homomorphism
LieDerivation + LieDerivation -- addition of Lie derivations
LieDerivation - LieDerivation -- subtraction of Lie derivations
LieDerivation @ LieElement -- formal application of a derivation to a Lie element
LieDerivation \ List -- apply a derivation to every element in a list
LieDerivation \\ List -- formal application of a derivation to every element in a list
LieDerivation LieDerivation -- Lie multiplication of ordinary derivations
LieDerivation LieElement -- apply a derivation
map(LieDerivation) -- get the map in the definition of a Lie derivation
Number LieDerivation -- multiplication of a number and a derivation
RingElement LieDerivation -- multiplication of a field element and a derivation
sign(LieDerivation) -- see sign -- get the sign of a homogeneous element
source(LieDerivation) -- get the source of a map
target(LieDerivation) -- get the target of a map
weight(LieDerivation) -- see weight -- get the weight of a homogeneous element

For the programmer

The object LieDerivation is a type, with ancestor classes HashTable < Thing.

LieDerivation -- the class of all Lie algebra derivations

Description

See also

Functions and methods returning an object of class LieDerivation :

Methods that use an object of class LieDerivation :

For the programmer