# ambient(LieAlgebra) -- get the ambient Lie algebra

## Synopsis

• Function: ambient
• Usage:
M = ambient L
• Inputs:
• 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