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# decompose(LieAlgebra) -- compute the ideal associated to an arrangement or matroid

## Synopsis

• Function: decompose
• Usage:
I = decompose(L)
• Inputs:
• L, an instance of the type LieAlgebra, the holonomy Lie algebra
• Outputs:

## Description

This gives the kernel of the Lie homomorphism from [$L,L$] to the direct sum of [$L_i,L_i$], where $L_i$ is the Lie subalgebra generated by the $i$th subset in the input for the holonomy Lie algebra $L$; see holonomyLocal. The ideal is generated by the basis elements in degree 3 of the form (x y z), where not all x,y,z belong to the same $L_i$. The ideal is zero if and only if $L$ decomposes into the direct sum of the local Lie subalgebras $L_i$ in degrees $\ge \ 2$.

 i1 : L=holonomy({{a0,a1,a2},{a0,a3,a4},{a1,a3,a5},{a2,a4,a5}}) o1 = L o1 : LieAlgebra i2 : I=decompose L o2 = I o2 : FGLieIdeal i3 : dims(1,4,I) o3 = {0, 0, 2, 9} o3 : List i4 : basis(3,I) o4 = {(a5 a4 a3), (a4 a5 a3)} o4 : List