next | previous | forward | backward | up | index | toc

# lieSubAlgebra -- make a Lie subalgebra

## Synopsis

• Usage:
S=lieSubAlgebra(gens)
• Inputs:
• gens, a list, a list of elements in a Lie algebra $L$
• Outputs:
• S, an instance of the type FGLieSubAlgebra, the subalgebra of $L$ generated by the list gens

## Description

The input should be a list of Lie elements in a Lie algebra $L$. The program adds generators for the subalgebra to make it invariant under the differential.

 i1 : F=lieAlgebra({a,b,c,r3,r4,r42}, Weights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}}, Signs=>{0,0,0,1,1,0},LastWeightHomological=>true) o1 = F o1 : LieAlgebra i2 : D=differentialLieAlgebra{0_F,0_F,0_F,a c,a a c,r4 - a r3} o2 = D o2 : LieAlgebra i3 : S=lieSubAlgebra{b c - a c,a b,b r4 - a r4} o3 = S o3 : FGLieSubAlgebra i4 : describe S o4 = generators => { - (a c) + (b c), - (b a), - (a r4) + (b r4), - (a a a lieAlgebra => D ------------------------------------------------------------------------ c) + (b a a c)} i5 : basis(5,S) o5 = {(a b a c) - (a b b c) - (b a a c) + (b a b c), (a a a c) - (b a a c), ------------------------------------------------------------------------ (a r4) - (b r4)} o5 : List i6 : d=differential D o6 = d o6 : LieDerivation i7 : d\S#gens o7 = {0, 0, - (a a a c) + (b a a c), 0} o7 : List i8 : (b c-a c) a b o8 = (a b a c) - (a b b c) - (b a a c) + (b a b c) o8 : D