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

- 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

## See also

- holonomy -- compute the holonomy Lie algebra associated to an arrangement or matroid
- decompose(LieAlgebra) -- compute the ideal associated to an arrangement or matroid
- Holonomy Lie algebras and symmetries

## Ways to use holonomyLocal:

- `holonomyLocal(ZZ,LieAlgebra)`

## For the programmer

The object holonomyLocal is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc2.m2:1584:0.