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# LieSubSpace @ LieSubSpace -- make the intersection of two Lie subspaces

## Synopsis

• Operator: @
• Usage:
S = A @ B
• Inputs:
• A, an instance of the type LieSubSpace, an instance of LieSubSpace
• B, an instance of the type LieSubSpace, an instance of LieSubSpace
• Outputs:
• S, an instance of the type LieSubSpace, an instance of LieSubSpace, the intersection of $A$ and $B$

## Description

If both $A$ and $B$ are instances of LieIdeal, then $S$ is of type LieIdeal. If both $A$ and $B$ are instances of LieSubAlgebra but not both of LieIdeal, then $S$ is of type LieSubAlgebra. Otherwise, $S$ is of type LieSubSpace.

 i1 : L = lieAlgebra{a,b,c} o1 = L o1 : LieAlgebra i2 : A=lieIdeal{a} o2 = A o2 : FGLieIdeal i3 : B=lieIdeal{b} o3 = B o3 : FGLieIdeal i4 : S=A@B o4 = S o4 : LieIdeal i5 : basis(3,S) o5 = {(a b a), (b b a), (c b a), (b c a)} o5 : List i6 : T=A+B o6 = T o6 : FGLieIdeal i7 : dims(1,3,L/T) o7 = {1, 0, 0} o7 : List i8 : dims(1,5,L/A@B) o8 = {3, 2, 4, 6, 12} o8 : List i9 : dims(1,5,L/A++L/B) o9 = {4, 2, 4, 6, 12} o9 : List