# holonomyLocal -- compute the Lie algebra for a local subalgebra of the holonomy Lie algebra

## Synopsis

• Usage:
L = holonomyLocal(i,H)
• Inputs:
• i, an integer, referring to the $i$th set, where the sets in the first input of holonomy are counted before the sets in the second input
• H, an instance of the type LieAlgebra, the holonomy Lie algebra
• Outputs:
• L, an instance of the type LieAlgebra, the Lie subalgebra

## Description

The generators in the $i$th set (beginning with $i=0$) in the inputs of holonomy generate a subalgebra of the holonomy Lie algebra $H$, and the output of holonomyLocal(i,H) is this Lie subalgebra. If the set is of size $k$, then the local Lie algebra is free on $k$ generators if the set belongs to the first input set, and it is free on $k-1$ generators in degrees $\ge 2$ if it belongs to the second input set.

 i1 : H=holonomy({{a1,a2},{a3,a4}},{{a1,a3,a5},{a2,a4,a5}}) o1 = H o1 : LieAlgebra i2 : describe holonomyLocal(1,H) o2 = generators => {a3, a4} Weights => {{1, 0}, {1, 0}} Signs => {0, 0} ideal => {} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i3 : describe holonomyLocal(2,H) o3 = generators => {a1, a3, a5} Weights => {{1, 0}, {1, 0}, {1, 0}} Signs => {0, 0, 0} ideal => {(a3 a1) - (a5 a3), (a5 a1) + (a5 a3)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0

## Ways to use holonomyLocal :

• "holonomyLocal(ZZ,LieAlgebra)"

## For the programmer

The object holonomyLocal is .