DerLie -- a Type for Lie algebra derivations

Description

This Type represents graded derivations d:M->L where M,L are graded (differential) Lie algebras and L is an M-module via f:M->L. If M=L and f is the identity, the set of elements of class DerLie is a Lie algebra with Lie multiplication multDerLie. However it is not of class LieAlgebra, if we do not have a finite presentation.

i1 : L=lieAlgebra({x,y},{},genSigns=>1)

o1 = L

o1 : LieAlgebra

i2 : M=lieAlgebra({a,b},{},genSigns=>0,genWeights=>{2,2})

o2 = M

o2 : LieAlgebra

i3 : f = mapLie(L,M,{[x,x],[]})

o3 = f

o3 : MapLie

i4 : d = derLie(f,{[x,x],[x,y]})

o4 = d

o4 : DerLie

i5 : peek f

o5 = MapLie{a => [x, x]   }
            b => []
            sourceLie => M
            targetLie => L

i6 : peek d

o6 = DerLie{a => [x, x]        }
            b => [x, y]
            maplie => f
            signDer => 0
            sourceLie => M
            targetLie => L
            weightDer => {0, 0}

i7 : evalDerLie(d,[a,b])

o7 = {{-1}, {[x, y, x, x]}}

o7 : List

Functions and methods returning an object of class DerLie :

derLie -- constructing a graded derivation
multDerLie -- defines the Lie multiplication of two derivations on a Lie algebra

Methods that use an object of class DerLie :

evalDerLie(DerLie,Array), see evalDerLie -- the value of a Lie derivation applied to an argument
evalDerLie(DerLie,List), see evalDerLie -- the value of a Lie derivation applied to an argument
invImageBasisLie(ZZ,DerLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
invImageBasisLie(ZZ,ZZ,DerLie,List), see invImageBasisLie -- computes a basis for the inverse image of a map or derivation
invImageLie(ZZ,DerLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
invImageLie(ZZ,ZZ,DerLie,List), see invImageLie -- computes the dimension for the inverse image of a map or derivation
multDerLie(DerLie,DerLie), see multDerLie -- defines the Lie multiplication of two derivations on a Lie algebra

For the programmer

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

DerLie -- a Type for Lie algebra derivations

Description

See also

Functions and methods returning an object of class DerLie :

Methods that use an object of class DerLie :

For the programmer