ambient(LieAlgebra) -- get the ambient Lie algebra

Function: ambient
Usage:

M = ambient L
Inputs:
- L, an instance of the type LieAlgebra,
Outputs:
- M, an instance of the type LieAlgebra, a Lie algebra of which $L$ is a quotient

Description

Relations in $L$ are elements in ambient(L). When a quotient Lie algebra Q=L/I is constructed, where $I$ is a list, then the elements in $I$ must be of type $L$, but the program converts the relations so that they have type ambient(L) instead. This may be seen by looking at ideal(Q). If $I$ is of type LieIdeal, then the value of the expression ambient(Q) is L, and the value of the expression ideal(Q) is I.

i1 : M = lieAlgebra{a,b,c}

o1 = M

o1 : LieAlgebra

i2 : L = M/{a b}

o2 = L

o2 : LieAlgebra

i3 : a c

o3 =  - (c a)

o3 : L

i4 : Q = L/{a c}

o4 = Q

o4 : LieAlgebra

i5 : ideal(Q)

o5 = { - (b a),  - (c a)}

o5 : List

i6 : class\oo

o6 = {M, M}

o6 : List

i7 : f=map(Q,L)

o7 = f

o7 : LieAlgebraMap

i8 : I = kernel f

o8 = I

o8 : LieIdeal

i9 : R = L/I

o9 = R

o9 : LieAlgebra

i10 : ambient R

o10 = L

o10 : LieAlgebra

i11 : ideal R

o11 = I

o11 : LieIdeal

Ways to use this method:

ambient(LieAlgebra) -- get the ambient Lie algebra

The source of this document is in GradedLieAlgebras/doc.m2:3099:0.